Infinite Sequences and Series

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Inﬁnite Sequences 11 and Series

In the last section of this chapter you are asked to use a series to derive a formula for the velocity of an ocean wave. © Epic Stock / Shutterstock

Infinite sequences and series were introduced briefly in A Preview of Calculus in connection with Zeno’s paradoxes and the decimal representation of numbers. Their importance in calculus stems from Newton’s idea of representing functions as sums of infinite series. For instance, in finding areas he often integrated a function by first expressing it as a series and then integrating each term of the series. We will pursue his 2 idea in Section 11.10 in order to integrate such functions as eϪx . (Recall that we have previously been unable to do this.) Many of the functions that arise in mathematical physics and chemistry, such as Bessel functions, are defined as sums of series, so it is important to be familiar with the basic concepts of convergence of infinite sequences and series. Physicists also use series in another way, as we will see in Section 11.11. In studying fields as diverse as optics, special relativity, and electromagnetism, they analyze phenomena by replacing a function with the first few terms in the series that represents it.

689

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690 CHAPTER 11 INFINITE SEQUENCES AND SERIES

11.1 Sequences

A sequence can be thought of as a list of numbers written in a deﬁnite order:

a1, a2, a3, a4, ..., an, ...

The number a1 is called the ﬁrst term, a2 is the second term, and in general an is the nth term. We will deal exclusively with inﬁnite sequences and so each term an will have a succes-

sor anϩ1. Notice that for every positive integer na there is a corresponding number n and so a sequence can be deﬁned as a function whose domain is the set of positive integers. But we

usually write an instead of the function notation f n for the value of the function at the number n.

NOTATION The sequence {a1 , a2 , a3 , . . .} is also denoted by

ϱ an or an n෇1

EXAMPLE 1 Some sequences can be deﬁned by giving a formula for the nth term. In the following examples we give three descriptions of the sequence: one by using the preceding notation, another by using the deﬁning formula, and a third by writing out the terms of the sequence. Notice that n doesn’t have to start at 1.

ϱ n ෇ n 1 2 3 4 n (a) ϩ an ϩ , , , , ..., ϩ , ... n 1 n෇1 n 1 2 3 4 5 n 1

Ϫ1nn ϩ 1 Ϫ1nn ϩ 1 2 3 4 5 Ϫ1nn ϩ 1 (b) a ෇ Ϫ , , Ϫ , , ..., , ... 3n n 3n 3 9 27 81 3n

s Ϫ ϱ ෇ s Ϫ ജ s s s Ϫ (c) { n 3 }n෇3 an n 3 , n 3 {0, 1, 2 , 3 , ..., n 3 , ...}

ϱ ␲ ␲ s ␲ n ෇ n ജ 3 1 n (d) cos an cos , n 0 1, , , 0, . . . , cos , ... 6 n෇0 6 2 2 6

v EXAMPLE 2 Find a formula for the general term an of the sequence

3 4 5 6 7 , Ϫ , , Ϫ , , ... 5 25 125 625 3125

assuming that the pattern of the ﬁrst few terms continues. SOLUTION We are given that

3 4 5 6 7 a ෇ a ෇ Ϫ a ෇ a ෇ Ϫ a ෇ 1 5 2 25 3 125 4 625 5 3125

Notice that the numerators of these fractions start with 3 and increase by 1 whenever we go to the next term. The second term has numerator 4, the third term has numerator 5; in general, the nn th term will have numerator ϩ 2 . The denominators are the powers of 5,

SECTION 11.1 SEQUENCES 691

n so an has denominator 5 . The signs of the terms are alternately positive and negative, so we need to multiply by a power of Ϫ1 . In Example 1(b) the factor Ϫ1 n meant we started with a negative term. Here we want to start with a positive term and so we use Ϫ1 nϪ1 or Ϫ1 nϩ1 . Therefore

n ϩ 2 a ෇ Ϫ1 nϪ1 n 5 n

EXAMPLE 3 Here are some sequences that don’t have a simple deﬁning equation.

(a) The sequence pn , where pn is the population of the world as of January 1 in the year n.

(b) If we let an be the digit in the nth decimal place of the number e , then an is a well- deﬁned sequence whose ﬁrst few terms are

7, 1, 8, 2, 8, 1, 8, 2, 8, 4, 5, . . .

f1 ෇ 1 f2 ෇ 1 fn ෇ fnϪ1 ϩ fnϪ2 n ജ 3

Each term is the sum of the two preceding terms. The ﬁrst few terms are

1, 1, 2, 3, 5, 8, 13, 21, . . .

This sequence arose when the 13th-century Italian mathematician known as Fibonacci solved a problem concerning the breeding of rabbits (see Exercise 83).

a¢ A sequence such as the one in Example 1(a), an ෇ nn ϩ 1, can be pictured either by a¡ a™ a£ plotting its terms on a number line, as in Figure 1, or by plotting its graph, as in Figure 2. 011 Note that, since a sequence is a function whose domain is the set of positive integers, its 2 graph consists of isolated points with coordinates FIGURE 1 1, a12, a2 3, a3 . . . n, an . . .

an From Figure 1 or Figure 2 it appears that the terms of the sequence an ෇ nn ϩ 1 are approaching 1 as n becomes large. In fact, the difference 1 n 1 7 1 Ϫ ෇ a¶= 8 n ϩ 1 n ϩ 1

0 1 2 3 4 5 6 7 n can be made as small as we like by taking n sufﬁciently large. We indicate this by writing n lim ෇ 1 FIGURE 2 n l ϱ n ϩ 1

In general, the notation

lim an ෇ L n l ϱ

means that the terms of the sequence an approach Ln as becomes large. Notice that the following definition of the limit of a sequence is very similar to the definition of a limit of a function at infinity given in Section 2.6.

692 CHAPTER 11 INFINITE SEQUENCES AND SERIES

1 Deﬁnition A sequence an has the limit L and we write

lim an ෇ L or an l L as n l ϱ n l ϱ

if we can make the terms an as close to Ln as we like by taking sufﬁciently large. If limn l ϱ an exists, we say the sequence converges (or is convergent). Otherwise, we say the sequence diverges (or is divergent).

Figure 3 illustrates Deﬁnition 1 by showing the graphs of two sequences that have the limit L.

an an

L L FIGURE 3 Graphs of two sequences with 0 n 0 n lim an=L n `

A more precise version of Deﬁnition 1 is as follows.

2 Deﬁnition A sequence an has the limit L and we write

lim an ෇ L or an l L as n l ϱ n l ϱ Compare this deﬁnition with Deﬁnition 2.6.7. if for every ␧Ͼ0 there is a corresponding integer N such that

if n Ͼ N then an Ϫ L Ͻ ␧

Deﬁnition 2 is illustrated by Figure 4, in which the terms a1 , a2 , a3 , . . . are plotted on a number line. No matter how small an interval L Ϫ␧, L ϩ␧ is chosen, there exists an N

such that all terms of the sequence from aNϩ1 onward must lie in that interval.

a¡ a£ a™ aˆ aN+1 aN+2 a˜ a∞aß a¢ a¶ FIGURE 4 0 L-∑ L L+∑

Another illustration of Deﬁnition 2 is given in Figure 5. The points on the graph of an must lie between the horizontal lines y ෇ L ϩ␧ and y ෇ L Ϫ␧ if n Ͼ N . This picture must be valid no matter how small␧ is chosen, but usually a smaller ␧ requires a largerN .

y y=L+∑ L y=L-∑

0 n FIGURE 5 1 2 3 4 N

SECTION 11.1 SEQUENCES 693

If you compare Deﬁnition 2 with Deﬁnition 2.6.7 you will see that the only difference

between limn l ϱ an ෇ L and limx l ϱ f x ෇ Lnis that is required to be an integer. Thus we have the following theorem, which is illustrated by Figure 6.

3 Theorem If limx l ϱ f x ෇ Lfand n ෇ an when n is an integer, then limn l ϱ an ෇ L.

y y=ƒ

0 x FIGURE 6 1 2 3 4

r ෇ Ͼ In particular, since we know that limx l ϱ 1 x 0 when r 0 (Theorem 2.6.5), we have 1 4 ෇ Ͼ lim r 0 if r 0 n l ϱ n

If an becomes large as n becomes large, we use the notation lim n l ϱ an ෇ ϱ. The following precise deﬁnition is similar to Deﬁnition 2.6.9.

5 Deﬁnition limn l ϱ an ෇ ϱ means that for every positive number M there is an integer N such that

if n Ͼ Na then n Ͼ M

If lim n l ϱ an ෇ ϱ, then the sequence an is divergent but in a special way. We say that an diverges to ϱ . The Limit Laws given in Section 2.3 also hold for the limits of sequences and their proofs are similar.

Limit Laws for Sequences If an and bn are convergent sequences and c is a constant, then

lim an ϩ bn ෇ lim an ϩ lim bn n l ϱ n l ϱ n l ϱ

lim an Ϫ bn ෇ lim an Ϫ lim bn n l ϱ n l ϱ n l ϱ

lim can ෇ c lim an lim c ෇ c n l ϱ n l ϱ n l ϱ lim an bn ෇ lim an ؒ lim bn n l ϱ n l ϱ n l ϱ

lim an an n l ϱ lim ෇ if lim bn 0 l ϱ l ϱ n bn lim bn n n l ϱ

p p lim an ෇ lim an if p Ͼ 0 and an Ͼ 0 n l ϱ n l ϱ

694 CHAPTER 11 INFINITE SEQUENCES AND SERIES

The Squeeze Theorem can also be adapted for sequences as follows (see Figure 7).

Squeeze Theorem for Sequences If an ഛ bn ഛ cn for n ജ n0 and lim an ෇ lim cn ෇ L, then lim bn ෇ L . n l ϱ n l ϱ n l ϱ

Another useful fact about limits of sequences is given by the following theorem, whose c n proof is left as Exercise 87.

bn 6 Theorem If lim an ෇ 0, then lim an ෇ 0 . n l ϱ n l ϱ an

0 n n EXAMPLE 4 Find lim . n l ϱ ϩ FIGURE 7 n 1

The sequence bn is squeezed SOLUTION The method is similar to the one we used in Section 2.6: Divide numerator between the sequences an and denominator by the highest power of n that occurs in the denominator and then use and cn . the Limit Laws. lim 1 n 1 l ϱ lim ෇ lim ෇ n n l ϱ n ϩ 1 n l ϱ 1 1 1 ϩ lim 1 ϩ lim n n l ϱ n l ϱ n This shows that the guess we made earlier 1 ෇ ෇ 1 from Figures 1 and 2 was correct. 1 ϩ 0

Here we used Equation 4 with r ෇ 1 .

n EXAMPLE 5 Is the sequence an ෇ convergent or divergent? s10 ϩ n SOLUTION As in Example 4, we divide numerator and denominator by n :

n 1 lim ෇ lim ෇ ϱ n l ϱ s10 ϩ n n l ϱ 10 ϩ 1 n 2 n

because the numerator is constant and the denominator approaches 0 . So an is divergent. ln n EXAMPLE 6 Calculate lim . n l ϱ n

SOLUTION Notice that both numerator and denominator approach inﬁnity as n l ϱ . We can’t apply l’Hospital’s Rule directly because it applies not to sequences but to functions of a real variable. However, we can apply l’Hospital’s Rule to the related function f x ෇ ln xx and obtain

ln x 1x lim ෇ lim ෇ 0 x l ϱ x x l ϱ 1

Therefore, by Theorem 3, we have ln n lim ෇ 0 n l ϱ n

SECTION 11.1 SEQUENCES 695

an n EXAMPLE 7 Determine whether the sequence an ෇ Ϫ1 is convergent or divergent. 1 SOLUTION If we write out the terms of the sequence, we obtain Ϫ Ϫ Ϫ Ϫ 0 1 2 3 4 n 1, 1, 1, 1, 1, 1, 1, ... _1 The graph of this sequence is shown in Figure 8. Since the terms oscillate between 1 and n Ϫ1 inﬁnitely often, an does not approach any number. Thus lim n l ϱ Ϫ1 does not exist; Ϫ n FIGURE 8 that is, the sequence 1 is divergent.

Ϫ1n The graph of the sequence in Example 8 is EXAMPLE 8 Evaluate lim if it exists. shown in Figure 9 and supports our answer. n l ϱ n SOLUTION We ﬁrst calculate the limit of the absolute value: an 1 Ϫ1n 1 lim ෇ lim ෇ 0 n l ϱ n n l ϱ n Therefore, by Theorem 6, n 0 n Ϫ1 1 lim ෇ 0 n l ϱ n

The following theorem says that if we apply a continuous function to the terms of a con- _1 vergent sequence, the result is also convergent. The proof is left as Exercise 88.

FIGURE 9 7 Theorem If lim an ෇ LfLand the function is continuous at , then n l ϱ

lim f an ෇ f L n l ϱ

EXAMPLE 9 Find lim sin␲n. n l ϱ

SOLUTION Because the sine function is continuous at 0 , Theorem 7 enables us to write Creating Graphs of Sequences Some computer algebra systems have special lim sin␲n ෇ sinlim ␲n ෇ sin 0 ෇ 0 commands that enable us to create sequences n l ϱ n l ϱ and graph them directly. With most graphing n calculators, however, sequences can be v EXAMPLE 10 Discuss the convergence of the sequence an ෇ n!n , where .graphed by using parametric equations. For n! ෇ 1 ؒ 2 ؒ 3 ؒ иии ؒ n instance, the sequence in Example 10 can be graphed by entering the parametric equations SOLUTION Both numerator and denominator approach inﬁnity as n l ϱ but here we x ෇ ty෇ t!t t have no corresponding function for use with l’Hospital’s Rule (x! is not deﬁned when x is not an integer). Let’s write out a few terms to get a feeling for what happens to an and graphing in dot mode, starting with t ෇ 1 as n gets large: and setting the t -step equal to . The result is ؒ ؒ ؒ 1 shown in Figure 10. ෇ ෇ 1 2 ෇ 1 2 3 ؒ ؒ a1 1 a2 ؒ a3 1 2 2 3 3 3 иии ؒ n ؒ 3 ؒ 2 ؒ 1 8 a ෇ n n ؒ n ؒ n ؒ иии ؒ n

It appears from these expressions and the graph in Figure 10 that the terms are decreasing and perhaps approach 0. To conﬁrm this, observe from Equation 8 that ؒ иии ؒ ؒ 10 0 ෇ 1 2 3 n an FIGURE 10 n n ؒ n ؒ иии ؒ n

696 CHAPTER 11 INFINITE SEQUENCES AND SERIES

Notice that the expression in parentheses is at most 1 because the numerator is less than (or equal to) the denominator. So 1 0 Ͻ a ഛ n n

We know that 1n l 0 as n l ϱ . Therefore an l 0 as n l ϱ by the Squeeze Theorem.

v EXAMPLE 11 For what values of r is the sequence r n convergent? SOLUTION We know from Section 2.6 and the graphs of the exponential functions in x x Section 1.5 that limx l ϱ a ෇ ϱ for a Ͼ 1 and limx l ϱ a ෇ 00for Ͻ a Ͻ 1 . Therefore, putting a ෇ r and using Theorem 3, we have

ϱ if r Ͼ 1 lim r n ෇ n l ϱ 0 if 0 Ͻ r Ͻ 1

It is obvious that lim 1n ෇ 1 and lim 0 n ෇ 0 n l ϱ n l ϱ

If Ϫ1 Ͻ r Ͻ 00, then Ͻ r Ͻ 1 , so

lim r n ෇ lim r n ෇ 0 n l ϱ n l ϱ

n n and therefore lim n l ϱ r ෇ 0 by Theorem 6. If r ഛϪ1 , then r diverges as in Example 7. Figure 11 shows the graphs for various values of rr . (The case ෇ Ϫ1 is shown in Figure 8.)

an an

r>1

1 1 _1

0 n r<_1 FIGURE 11 1 0

The results of Example 11 are summarized for future use as follows.

9 The sequence r n is convergent if Ϫ1 Ͻ r ഛ 1 and divergent for all other values of r . 0 if Ϫ1 Ͻ r Ͻ 1 lim r n ෇ n l ϱ 1 if r ෇ 1

10 Definition A sequence an is called increasing if an Ͻ anϩ1 for all n ജ 1 , that is, a1 Ͻ a2 Ͻ a3 Ͻ иии. It is called decreasing if an Ͼ anϩ1 for all n ജ 1 . A sequence is monotonic if it is either increasing or decreasing.

SECTION 11.1 SEQUENCES 697

3 EXAMPLE 12 The sequence is decreasing because n ϩ 5

The right side is smaller because it has a 3 Ͼ 3 ෇ 3 larger denominator. n ϩ 5 n ϩ 1 ϩ 5 n ϩ 6

and so an Ͼ anϩ1 for all n ജ 1 .

n EXAMPLE 13 Show that the sequence a ෇ is decreasing. n n2 ϩ 1

SOLUTION 1 We must show that anϩ1 Ͻ an , that is,

n ϩ 1 n Ͻ n ϩ 12 ϩ 1 n2 ϩ 1

This inequality is equivalent to the one we get by cross-multiplication:

n ϩ 1 n Ͻ &? n ϩ 1n2 ϩ 1 Ͻ nn ϩ 12 ϩ 1 n ϩ 12 ϩ 1 n2 ϩ 1 &? n3 ϩ n2 ϩ n ϩ 1 Ͻ n3 ϩ 2n2 ϩ 2n

&? 1 Ͻ n2 ϩ n

2 Since n ജ 1 , we know that the inequality n ϩ n Ͼ 1 is true. Therefore anϩ1 Ͻ an and so an is decreasing. x SOLUTION 2 Consider the function f x ෇ : x 2 ϩ 1

x 2 ϩ 1 Ϫ 2x 2 1 Ϫ x 2 f Јx ෇ ෇ Ͻ 0 whenever x2 Ͼ 1 x 2 ϩ 12 x 2 ϩ 12

Thus f is decreasing on 1, ϱ and so f n Ͼ f n ϩ 1. Therefore an is decreasing.

11 Deﬁnition A sequence an is bounded above if there is a number M such that

an ഛ M for all n ജ 1

It is bounded below if there is a number m such that

m ഛ an for all n ജ 1

If it is bounded above and below, then an is a bounded sequence.

For instance, the sequence an ෇ n is bounded below an Ͼ 0 but not above. The sequence an ෇ nn ϩ 1 is bounded because 0 Ͻ an Ͻ 1 for all n . We know that not every bounded sequence is convergent [for instance, the sequence n an ෇ Ϫ1 satisﬁes Ϫ1 ഛ an ഛ 1 but is divergent from Example 7] and not every mono-

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698 CHAPTER 11 INFINITE SEQUENCES AND SERIES

tonic sequence is convergent ͑an ෇ n l ϱ͒. But if a sequence is both bounded and monotonic, then it must be convergent. This fact is proved as Theorem 12, but intuitively

you can understand why it is true by looking at Figure 12. If ͕an ͖ is increasing and an ഛ M for all nL , then the terms are forced to crowd together and approach some number .

an M

0 FIGURE 12 1 2 3 n

The proof of Theorem 12 is based on the Completeness Axiom for the set ޒ of real numbers, which says that if SM is a nonempty set of real numbers that has an upper bound (x ഛ M for all xS in ), then S has a least upper bound b. (This means that b is an upper bound for SM , but if is any other upper bound, then bഛ M .) The Completeness Axiom is an expression of the fact that there is no gap or hole in the real number line.

12 Monotonic Sequence Theorem Every bounded, monotonic sequence is convergent.

PROOF Suppose ͕an ͖͕ is an increasing sequence. Since an ͖ is bounded, the set S ෇ ͕an Խ n ജ 1͖ has an upper bound. By the Completeness Axiom it has a least upper bound L . Given ␧Ͼ0 , L Ϫ␧ is not an upper bound for SL (since is the least upper bound). Therefore

aN Ͼ L Ϫ␧ for some integer N

But the sequence is increasing so an ജ aN for every n Ͼ Nn . Thus if Ͼ N , we have

an Ͼ L Ϫ␧

so 0 ഛ L Ϫ an Ͻ ␧

since an ഛ L. Thus

Խ L Ϫ an Խ Ͻ ␧ whenever n Ͼ N

so lim n l ϱ an ෇ L. A similar proof (using the greatest lower bound) works if ͕an ͖ is decreasing.

The proof of Theorem 12 shows that a sequence that is increasing and bounded above is convergent. (Likewise, a decreasing sequence that is bounded below is convergent.) This fact is used many times in dealing with inﬁnite series.

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SECTION 11.1 SEQUENCES 699

EXAMPLE 14 Investigate the sequence ͕an ͖ deﬁned by the recurrence relation

෇ ෇ 1 ͑ ϩ ͒ ෇ a1 2 anϩ1 2 an 6 for n 1, 2, 3, . . .

SOLUTION We begin by computing the ﬁrst several terms:

෇ ෇ 1 ͑ ϩ ͒ ෇ ෇ 1 ͑ ϩ ͒ ෇ a1 2 a2 2 2 6 4 a3 2 4 6 5

෇ 1 ͑ ϩ ͒ ෇ ෇ ෇ a4 2 5 6 5.5 a5 5.75 a6 5.875

a7 ෇ 5.9375 a8 ෇ 5.96875 a9 ෇ 5.984375 Mathematical induction is often used in dealing with recursive sequences. See page 76 for These initial terms suggest that the sequence is increasing and the terms are approaching a discussion of the Principle of Mathematical 6. To conﬁrm that the sequence is increasing, we use mathematical induction to show Induction. that anϩ1 Ͼ an for all n ജ 1 . This is true for n ෇ 1 because a2 ෇ 4 Ͼ a1. If we assume that it is true for n ෇ k , then we have

akϩ1 Ͼ ak

so akϩ1 ϩ 6 Ͼ ak ϩ 6

1 ͑ ϩ ͒ Ͼ 1 ͑ ϩ ͒ and 2 akϩ1 6 2 ak 6

Thus akϩ2 Ͼ akϩ1

We have deduced that anϩ1 Ͼ an is true for n ෇ k ϩ 1 . Therefore the inequality is true for all n by induction.

Next we verify that ͕an ͖ is bounded by showing that an Ͻ 6 for all n . (Since the

sequence is increasing, we already know that it has a lower bound: an ജ a1 ෇ 2 for all na .) We know that 1 Ͻ 6 , so the assertion is true for n ෇ 1 . Suppose it is true for n ෇ k. Then

ak Ͻ 6

so ak ϩ 6 Ͻ 12 1 ͑ ϩ ͒ Ͻ 1 ͑ ͒ ෇ and 2 ak 6 2 12 6

Thus akϩ1 Ͻ 6

This shows, by mathematical induction, that an Ͻ 6 for all n . Since the sequence ͕an ͖ is increasing and bounded, Theorem 12 guarantees that it has a limit. The theorem doesn’t tell us what the value of the limit is. But now that we know

L ෇ limn l ϱ an exists, we can use the given recurrence relation to write

෇ 1 ͑ ϩ ͒ ෇ 1 ϩ ෇ 1 ͑ ϩ ͒ lim anϩ1 lim 2 an 6 2 lim an 6 2 L 6 n l ϱ n l ϱ ( n l ϱ )

A proof of this fact is requested in Exercise 70. Since an l La , it follows that nϩ1 l Ln too (as l ϱ , n ϩ 1 l ϱ also). So we have

෇ 1 ͑ ϩ ͒ L 2 L 6

Solving this equation for LL , we get ෇ 6 , as we predicted.

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700 CHAPTER 11 INFINITE SEQUENCES AND SERIES

11.1 Exercises

1. (a) What is a sequence? 23–56 Determine whether the sequence converges or diverges.

(b) What does it mean to say that limn l ϱ an ෇ 8? If it converges, find the limit. (c) What does it mean to say that lim l ϱ a ෇ ϱ? n n n3 23. a ෇ 1 Ϫ ͑0.2͒n 24. a ෇ n n 3 ϩ 2. (a) What is a convergent sequence? Give two examples. n 1 (b) What is a divergent sequence? Give two examples. 3 ϩ 5n2 n3 25. a ෇ 26. a ෇ n ϩ 2 n ϩ 3–12 List the ﬁrst ﬁve terms of the sequence. n n n 1

n nϩ2 2n 3 1͞n 3 3. a ෇ 4. a ෇ 27. an ෇ e 28. an ෇ n n 2 ϩ 1 n 1 ϩ 2n 5n

͑Ϫ1͒nϪ1 n␲ 5. ෇ 6. ෇ 2n␲ n ϩ 1 an n an cos 29. a ෇ tanͩ ͪ 30. a ෇ ͱ 5 2 n 1 ϩ 8n n 9n ϩ 1 1 ͑Ϫ1͒nn 7. ෇ 8. ෇ n2 an an 31. ෇ 32. ෇ 2n͑͞nϩ2͒ ͑n ϩ 1͒! n! ϩ 1 an an e sn3 ϩ 4n

9. a ෇ 1, a ϩ ෇ 5a Ϫ 3 1 n 1 n ͑Ϫ1͒n ͑Ϫ1͒nϩ1n 33. an ෇ 34. an ෇ an 2sn n ϩ sn 10. a ෇ 6, a ϩ ෇ 1 n 1 n 35. an ෇ cos͑n͞2͒ 36. an ෇ cos͑2͞n͒ a 11. ෇ ෇ n a1 2, anϩ1 ϩ 1 an ͑2n Ϫ 1͒! ln n 37. ͭ ͮͭ38. ͮ ͑2n ϩ 1͒! ln 2n 12. a1 ෇ 2, a2 ෇ 1 , anϩ1 ෇ an Ϫ anϪ1

e n ϩ e Ϫn tanϪ1n 39. ͭ ͮ 40. a ෇ e 2n Ϫ 1 n n 13–18 Find a formula for the general term an of the sequence, assuming that the pattern of the ﬁrst few terms continues. 2 Ϫn 41. ͕n e ͖ 42. an ෇ ln͑n ϩ 1͒ Ϫ ln n 13. 1 1 1 1 {1, 3 , 5 , 7 , 9 , ...} cos2n 14. ͕1, Ϫ1 , 1 , Ϫ 1 , 1 , ...͖ 43. a ෇ 44. a ෇ sn 21ϩ3n 3 9 27 81 n 2n n 15. ͕Ϫ Ϫ4 8 Ϫ16 ͖ 3, 2, 3 , 9 , 27 , ... Ϫn 45. an ෇ n sin͑1͞n͒ 46. an ෇ 2 cos n␲ 16. ͕5, 8, 11, 14, 17, . . .͖ n 2 sin 2n 17. ͕ 1 Ϫ4 9 Ϫ16 25 ͖ 47. ෇ ͩ ϩ ͪ 48. ෇ , , , , , ... an 1 an 2 3 4 5 6 n 1 ϩ sn 18. ͕1, 0, Ϫ1, 0, 1, 0, Ϫ1, 0, . . .͖ 2 2 49. an ෇ ln͑2n ϩ 1͒ Ϫ ln͑n ϩ 1͒

19–22 Calculate, to four decimal places, the first ten terms of the ͑ln n͒2 sequence and use them to plot the graph of the sequence by hand. 50. an ෇ n Does the sequence appear to have a limit? If so, calculate it. If not, explain why. 51. an ෇ arctan͑ln n͒ 3n ͑Ϫ1͒n 19. ෇ 20. ෇ ϩ an an 2 52. a ෇ n Ϫ sn ϩ 1 sn ϩ 3 1 ϩ 6n n n

n 53. ͕ ͖ n 10 0, 1, 0, 0, 1, 0, 0, 0, 1, . . . 21. ෇ ϩ Ϫ1 22. ෇ ϩ an 1 ( 2 ) an 1 n 9 54. 1 1 1 1 1 1 1 1 {1 , 3 , 2 , 4 , 3 , 5 , 4 , 6 , ...}

; Graphing calculator or computer required 1. Homework Hints available at stewartcalculus.com

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CHAPTER 11.1 SEQUENCES 701

n! ͑Ϫ3͒n 68. Find the ﬁrst 40 terms of the sequence deﬁned by 55. a ෇ 56. a ෇ n 2n n n! 1 2 an if an is an even number ϩ ෇ ͭ an 1 3an ϩ 1 if an is an odd number ; 57–63 Use a graph of the sequence to decide whether the ෇ ෇ sequence is convergent or divergent. If the sequence is conver- and a1 11 . Do the same if a1 25 . Make a conjecture gent, guess the value of the limit from the graph and then prove about this type of sequence. your guess. (See the margin note on page 695 for advice on 69. For what values of r is the sequence ͕nr n ͖ convergent? graphing sequences.) 70. (a) If ͕an ͖ is convergent, show that n 57. an ෇ 1 ϩ ͑Ϫ2͞e͒ 58. an ෇ sn sin(␲͞sn )

lim anϩ1 ෇ lim an l ϱ l ϱ 3 ϩ 2n2 n n 59. a ෇ ͱ 60. a ෇ sn 3n ϩ 5n n 8n2 ϩ n n (b) A sequence ͕an ͖ is deﬁned by a1 ෇ 1 and

anϩ1 ෇ 1͑͞1 ϩ an ͒ for n ജ 1 . Assuming that ͕an ͖ is n2 cos n 61. a ෇ convergent, ﬁnd its limit. n 1 ϩ n2 71. Suppose you know that ͕an ͖ is a decreasing sequence and иии ؒ ͑2n Ϫ 1͒ all its terms lie between the numbers 5 and 8. Explain why ؒ 5 ؒ 3 ؒ 1 62. a ෇ n n! the sequence has a limit. What can you say about the value of the limit? иии ؒ ͑2n Ϫ 1͒ ؒ 5 ؒ 3 ؒ 1 63. a ෇ 72–78 Determine whether the sequence is increasing, decreasing, n ͑ ͒n 2n or not monotonic. Is the sequence bounded?

nϩ1 72. an ෇ ͑Ϫ2͒ 64. (a) Determine whether the sequence deﬁned as follows is 1 2n Ϫ 3 73. a ෇ 74. a ෇ convergent or divergent: n 2n ϩ 3 n 3n ϩ 4 75. ෇ ͑Ϫ ͒n 76. ෇ Ϫn a1 ෇ 1 anϩ1 ෇ 4 Ϫ an for n ജ 1 an n 1 an ne ෇ n 1 (b) What happens if the ﬁrst term is a1 2 ? 77. a ෇ 78. a ෇ n ϩ n n 2 ϩ 1 n n 65. If $1000 is invested at 6% interest, compounded annually, n then after na years the investment is worth n ෇ 1000͑1.06͒ 79. dollars. Find the limit of the sequence

(a) Find the ﬁrst ﬁve terms of the sequence ͕an ͖ . (b) Is the sequence convergent or divergent? Explain. {s2 , s2s2 , s2s2s2 , ...}

66. If you deposit $100 at the end of every month into an 80. A sequence ͕an ͖ is given by a1 ෇ s2 , anϩ1 ෇ s2 ϩ an .

account that pays 3% interest per year compounded monthly, (a) By induction or otherwise, show that ͕an ͖ is increasing the amount of interest accumulated after n months is given and bounded above by 3. Apply the Monotonic Sequence

by the sequence Theorem to show that limn l ϱ an exists. (b) Find lim l ϱ a . 1.0025n Ϫ 1 n n I ෇ 100ͩ Ϫ nͪ n 0.0025 81. Show that the sequence deﬁned by 1 (a) Find the ﬁrst six terms of the sequence. a1 ෇ 1 anϩ1 ෇ 3 Ϫ (b) How much interest will you have earned after two years? an

is increasing and an Ͻ 3 for all n . Deduce that ͕an ͖ is conver- 67. A fish farmer has 5000 catfish in his pond. The number of gent and find its limit. catfish increases by 8% per month and the farmer harvests 82. 300 catfish per month. Show that the sequence defined by (a) Show that the catfish population P after n months is n ෇ ෇ 1 given recursively by a1 2 anϩ1 3 Ϫ an

Pn ෇ 1.08PnϪ1 Ϫ 300 P0 ෇ 5000 satisfies 0 Ͻ an ഛ 2 and is decreasing. Deduce that the (b) How many catfish are in the pond after six months? sequence is convergent and find its limit.

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702 CHAPTER 11 INFINITE SEQUENCES AND SERIES

83. (a) Fibonacci posed the following problem: Suppose that 91. Let ab and be positive numbers with aϾ ba . Let 1 be their

rabbits live forever and that every month each pair arithmetic mean and b1 their geometric mean: produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many a ϩ b a1 ෇ b1 ෇ sab pairs of rabbits will we have in the nth month? Show that 2 the answer is f , where ͕ f ͖ is the Fibonacci sequence n n Repeat this process so that, in general, deﬁned in Example 3(c).

(b) Let an ෇ fnϩ1͞fn and show that anϪ1 ෇ 1 ϩ 1͞anϪ2 . an ϩ bn Assuming that ͕a ͖ is convergent, find its limit. anϩ1 ෇ bnϩ1 ෇ san bn n 2 84. ෇ ෇ ͑ ͒ ෇ ͑ ͒ ෇ ͑ ͑ ͒͒ (a) Let a1 aa , 2 f a , a3 f a2 f f a , . . . , (a) Use mathematical induction to show that anϩ1 ෇ f ͑an ͒, where f is a continuous function. If l ϱ ෇ ͑ ͒ ෇ limn an Lf, show that L L . an Ͼ anϩ1 Ͼ bnϩ1 Ͼ bn (b) Illustrate part (a) by taking f ͑x͒ ෇ cos xa, ෇ 1 , and estimating the value of L to five decimal places. (b) Deduce that both ͕an ͖͕ and bn ͖ are convergent. (c) Show that lim l ϱ a ෇ lim l ϱ b . Gauss called the ; 85. (a) Use a graph to guess the value of the limit n n n n common value of these limits the arithmetic-geometric n5 mean of the numbers ab and . lim nl ϱ n! 92. (a) Show that if lim n l ϱ a2n ෇ L and lim n l ϱ a2nϩ1 ෇ L, ͕ ͖ ෇ (b) Use a graph of the sequence in part (a) to find the then an is convergent and lim n l ϱ an L. ෇ smallest values of N that correspond to ␧ ෇ 0.1 and (b) If a1 1 and ␧ ෇ 0.001 in Definition 2. 1 anϩ1 ෇ 1 ϩ n 1 ϩ an 86. Use Definition 2 directly to prove that lim n l ϱ r ෇ 0 when Խ r Խ Ͻ 1. find the first eight terms of the sequence ͕an ͖ . Then use ෇ s 87. Prove Theorem 6. part (a) to show that lim n l ϱ an 2 . This gives the [Hint: Use either Definition 2 or the Squeeze Theorem.] continued fraction expansion

88. 1 Prove Theorem 7. s2 ෇ 1 ϩ 1 89. ෇ ͕ ͖ 2 ϩ Prove that if limn l ϱ an 0 and bn is bounded, then 2 ϩиии limn l ϱ ͑an bn͒ ෇ 0. 93. The size of an undisturbed ﬁsh population has been modeled 1 n by the formula 90. ෇ ͩ ϩ ͪ Let an 1 . bpn n p ϩ ෇ n 1 a ϩ p (a) Show that if 0 ഛ a Ͻ b , then n

where pn is the ﬁsh population after n years and ab and are ϩ ϩ b n 1 Ϫ a n 1 positive constants that depend on the species and its environ- Ͻ ͑n ϩ 1͒b n b Ϫ a ment. Suppose that the population in year 0 is p0 Ͼ 0 . (a) Show that if ͕ pn͖ is convergent, then the only possible (b) Deduce that b n ͓͑n ϩ 1͒a Ϫ nb͔ Ͻ a nϩ1. values for its limit are 0 and b Ϫ a .

show that ͕an ͖ is increasing. (c) Use part (b) to show that if a Ͼ b , then limn l ϱ pn ෇ 0; (d) Use a ෇ 1 and b ෇ 1 ϩ 1͑͞2n͒ in part (b) to show that in other words, the population dies out.

a2n Ͻ 4. (d) Now assume that a Ͻ bp . Show that if 0 Ͻ b Ϫ a , then

(e) Use parts (c) and (d) to show that an Ͻ 4 for all n . ͕ pn͖ is increasing and 0 Ͻ pn Ͻ b Ϫ a. Show also that n (f) Use Theorem 12 to show that lim n l ϱ ͑1 ϩ 1͞n͒ exists. if p 0 Ͼ b Ϫ a , then ͕ pn͖ is decreasing and pn Ͼ b Ϫ a .

(The limit is e . See Equation 3.6.6.) Deduce that if a Ͻ b , then limn l ϱ pn ෇ b Ϫ a.

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SECTION 11.2 SERIES 703

LABORATORY PROJECT CAS LOGISTIC SEQUENCES

A sequence that arises in ecology as a model for population growth is deﬁned by the logistic difference equation

pnϩ1 ෇ kpn͑1 Ϫ pn ͒

wherepn measures the size of the population of the nth generation of a single species. To keep

the numbers manageable,pn is a fraction of the maximal size of the population, so 0 ഛ pn ഛ 1 . Notice that the form of this equation is similar to the logistic differential equation in Section 9.4. The discrete model—with sequences instead of continuous functions—is preferable for modeling insect populations, where mating and death occur in a periodic fashion. An ecologist is interested in predicting the size of the population as time goes on, and asks these questions: Will it stabilize at a limiting value? Will it change in a cyclical fashion? Or will it exhibit random behavior? Write a program to compute the ﬁrst n terms of this sequence starting with an initial population

p0, where 0 Ͻ p0 Ͻ 1. Use this program to do the following. 1. ෇ 1 Calculate 20 or 30 terms of the sequence for p0 2 and for two values of k such that 1 Ͻ k Ͻ 3. Graph each sequence. Do the sequences appear to converge? Repeat for a dif-

ferent value of p0 between 0 and 1. Does the limit depend on the choice of p0 ? Does it depend on the choice of k ?

2. Calculate terms of the sequence for a value of k between 3 and 3.4 and plot them. What do you notice about the behavior of the terms?

3. Experiment with values of k between 3.4 and 3.5. What happens to the terms?

4. For values of k between 3.6 and 4, compute and plot at least 100 terms and comment on the

behavior of the sequence. What happens if you change p0 by 0.001? This type of behavior is called chaotic and is exhibited by insect populations under certain conditions.

CAS Computer algebra system required

11.2 Series

What do we mean when we express a number as an inﬁnite decimal? For instance, what does it mean to write

The current record is that ␲ has been computed ␲ ෇ 3.14159 26535 89793 23846 26433 83279 50288 . . . to 2,576,980,370,000 (more than two trillion) decimal places by T. Daisuke and his team. The convention behind our decimal notation is that any number can be written as an inﬁ- nite sum. Here it means that

1 4 1 5 9 2 6 5 ␲ ෇ 3 ϩ ϩ ϩ ϩ ϩ ϩ ϩ ϩ ϩ иии 10 10 2 10 3 10 4 10 5 10 6 107 10 8

where the three dots ͑иии͒ indicate that the sum continues forever, and the more terms we add, the closer we get to the actual value of ␲ .

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704 CHAPTER 11 INFINITE SEQUENCES AND SERIES

ϱ In general, if we try to add the terms of an inﬁnite sequence ͕an ͖n෇1 we get an expression of the form

1 a1 ϩ a2 ϩ a3 ϩиииϩan ϩиии

which is called an inﬁnite series (or just a series) and is denoted, for short, by the symbol

͚ an or ͚ an n෇1

Does it make sense to talk about the sum of infinitely many terms? It would be impossible to find a finite sum for the series

1 ϩ 2 ϩ 3 ϩ 4 ϩ 5 ϩиииϩn ϩиии

because if we start adding the terms we get the cumulative sums 1, 3, 6, 10, 15, 21, . . . and, after the nth term, we get n͑n ϩ 1͒͞2 , which becomes very large as n increases. However, if we start to add the terms of the series n Sum of first n terms 1 0.50000000 1 ϩ 1 ϩ 1 ϩ 1 ϩ 1 ϩ 1 ϩиииϩ 1 ϩиии 2 0.75000000 2 4 8 16 32 64 2n 3 0.87500000 4 0.93750000 1 3 7 15 31 63 Ϫ ͞ n 5 0.96875000 we get 2 , 4 , 8 , 16 , 32 , 64 , . . . , 1 1 2 , . . . . The table shows that as we add more and more 6 0.98437500 terms, these partial sums become closer and closer to 1. (See also Figure 11 in A Preview 7 0.99218750 of Calculus, page 6.) In fact, by adding sufficiently many terms of the series we can make 10 0.99902344 the partial sums as close as we like to 1. So it seems reasonable to say that the sum of this 15 0.99996948 infinite series is 1 and to write 20 0.99999905 25 0.99999997 ϱ 1 ෇ 1 ϩ 1 ϩ 1 ϩ 1 ϩиииϩ 1 ϩиии෇ ͚ n n 1 n෇1 2 2 4 8 16 2

We use a similar idea to determine whether or not a general series 1 has a sum. We consider the partial sums

s1 ෇ a1

s2 ෇ a1 ϩ a2

s3 ෇ a1 ϩ a2 ϩ a3

s4 ෇ a1 ϩ a2 ϩ a3 ϩ a4 and, in general,

sn ෇ a1 ϩ a2 ϩ a3 ϩиииϩan ෇ ͚ ai i෇1

These partial sums form a new sequence ͕sn ͖ , which may or may not have a limit. If lim l ϱ s ෇ s exists (as a ﬁnite number), then, as in the preceding example, we call it the n n ͸ sum of the inﬁnite series an .

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SECTION 11.2 SERIES 705

͸ϱ 2 Deﬁnition Given a series n෇1 an ෇ a1 ϩ a2 ϩ a3 ϩиии, let sn denote its nth partial sum: n

sn ෇ ͚ ai ෇ a1 ϩ a2 ϩиииϩan i෇1

If the sequence ͕s ͖ is convergent and lim l ϱ s ෇ s exists as a real number, then ͸ n n n the series an is called convergent and we write ϱ

a1 ϩ a2 ϩиииϩan ϩиии෇ s or ͚ an ෇ s n෇1

The number s is called the sum of the series. If the sequence ͕sn ͖ is divergent, then the series is called divergent.

Thus the sum of a series is the limit of the sequence of partial sums. So when we write ͸ϱ n෇1 an ෇ s, we mean that by adding sufficiently many terms of the series we can get as close as we like to the number s . Notice that Compare with the improper integral ϱ n ϱ t ͑ ͒ ෇ ͑ ͒ ͚ an ෇ lim ͚ ai y f x dx lim y f x dx l ϱ 1 t l ϱ 1 n෇1 n i෇1 To find this integral we integrate from 1 to t and then let t l ϱ . For a series, we sum from ͸ϱ EXAMPLE 1 Suppose we know that the sum of the first n terms of the series n෇1 an is 1 to n and then let n l ϱ . 2n s ෇ a ϩ a ϩ иии ϩ a ෇ n 1 2 n 3n ϩ 5

Then the sum of the series is the limit of the sequence ͕sn ͖ :

ϱ 2n 2 2 ͚ an ෇ lim sn ෇ lim ෇ lim ෇ ෇ n l ϱ n l ϱ 3n ϩ 5 n l ϱ 5 3 n 1 3 ϩ n

In Example 1 we were given an expression for the sum of the ﬁrst n terms, but it’s usually not easy to ﬁnd such an expression. In Example 2, however, we look at a famous series

for which we can ﬁnd an explicit formula for sn .

EXAMPLE 2 An important example of an inﬁnite series is the geometric series

ϱ a ϩ ar ϩ ar 2 ϩ ar 3 ϩиииϩar nϪ1 ϩиии෇ ͚ ar nϪ1 a 0 n෇1 Each term is obtained from the preceding one by multiplying it by the common ratio r. ෇ 1 ෇ 1 (We have already considered the special case where a 2 and r 2 on page 704.) If r ෇ 1 , then sn ෇ a ϩ a ϩиииϩa ෇ na l Ϯϱ. Since lim n l ϱ sn doesn’t exist, the geometric series diverges in this case. If r 1 , we have

2 nϪ1 sn ෇ a ϩ ar ϩ ar ϩиииϩar

2 nϪ1 n and rsn ෇ ar ϩ ar ϩиииϩar ϩ ar

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706 CHAPTER 11 INFINITE SEQUENCES AND SERIES

Figure 1 provides a geometric demonstration Subtracting these equations, we get of the result in Example 2. If the triangles are constructed as shown and is the sum of the n s sn Ϫ rsn ෇ a Ϫ ar series, then, by similar triangles, s a a a͑ Ϫ r n ͒ ෇ so s ෇ 1 Ϫ Ϫ 3 sn ෇ a a ar 1 r 1 Ϫ r

Ϫ Ͻ Ͻ n l l ϱ ar# If 1 r 1, we know from (11.1.9) that r 0 as n , so ar@ ͑ Ϫ n ͒ a 1 r a a n a lim sn ෇ lim ෇ Ϫ lim r ෇ ar@ n l ϱ n l ϱ 1 Ϫ r 1 Ϫ r 1 Ϫ r n l ϱ 1 Ϫ r ar Thus when Խ r Խ Ͻ 1 the geometric series is convergent and its sum is a͑͞1 Ϫ r͒ . a-ar ar If r ഛϪ1 or r Ͼ 1 , the sequence ͕r n ͖ is divergent by (11.1.9) and so, by Equation 3, s lim n l ϱ sn does not exist. Therefore the geometric series diverges in those cases.

We summarize the results of Example 2 as follows. a a

4 The geometric series

ϱ nϪ1 ෇ ϩ ϩ 2 ϩиии a ͚ ar a ar ar n෇1 FIGURE 1 is convergent if Խ r Խ Ͻ 1 and its sum is In words: The sum of a convergent geometric ϱ a series is ͚ nϪ1 ෇ Ͻ ar Ϫ Խ r Խ 1 first term n෇1 1 r Ϫ 1 common ratio If Խ r Խ ജ 1 , the geometric series is divergent.

v EXAMPLE 3 Find the sum of the geometric series

Ϫ 10 ϩ 20 Ϫ 40 ϩиии 5 3 9 27

෇ ෇ Ϫ 2 ෇ 2 Ͻ SOLUTION The ﬁrst term is a 5 and the common ratio is r 3 . Since Խ r Խ 3 1 , the series is convergent by 4 and its sum is

Ϫ 10 ϩ 20 Ϫ 40 ϩиии෇ 5 ෇ 5 ෇ 5 Ϫ Ϫ 2 5 3 3 9 27 1 ( 3 ) 3

What do we really mean when we say that the sn sum of the series in Example 3 is 3 ? Of course, n sn we can’t literally add an infinite number of 1 5.000000 terms, one by one. But, according to Defini - tion 2, the total sum is the limit of the 2 1.666667 sequence of partial sums. So, by taking the 3 3.888889 3 sum of sufficiently many terms, we can get as 4 2.407407 close as we like to the number 3 . The table 5 3.395062 shows the first ten partial sums and the sn 6 2.736626 graph in Figure 2 shows how the sequence of 7 3.175583 partial sums approaches 3 . 8 2.882945 0 20 n 9 3.078037 10 2.947975 FIGURE 2

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SECTION 11.2 SERIES 707

ϱ EXAMPLE 4 Is the series ͚ 22n31Ϫn convergent or divergent? n෇1 Ϫ SOLUTION Let’s rewrite the nth term of the series in the form ar n 1 : Another way to identify and is to write out ra ϱ ϱ ϱ n ϱ 4 Ϫ the ﬁrst few terms: 2n 1Ϫn ෇ ͑ 2͒n Ϫ͑nϪ1͒ ෇ ෇ 4 n 1 ͚ 2 3 ͚ 2 3 ͚ nϪ1 ͚ 4( 3 ) ϩ 16 ϩ 64 ϩиии n෇1 n෇1 n෇1 3 n෇1 4 3 9 ෇ ෇ 4 Ͼ We recognize this series as a geometric series with a 4 and r 3 . Since r 1 , the series diverges by 4 .

v EXAMPLE 5 Write the number 2.317 ෇ 2.3171717. . . as a ratio of integers.

SOLUTION 17 17 17 2.3171717. . . ෇ 2.3 ϩ ϩ ϩ ϩиии 103 105 107

After the ﬁrst term we have a geometric series with a ෇ 17͞103 and r ෇ 1͞102 . Therefore 17 17 103 1000 2.317 ෇ 2.3 ϩ ෇ 2.3 ϩ 1 99 1 Ϫ 102 100

෇ 23 ϩ 17 ෇ 1147 10 990 495

ϱ EXAMPLE 6 Find the sum of the series ͚ x n , where Խ x Խ Ͻ 1. n෇0 SOLUTION Notice that this series starts with n ෇ 0 and so the ﬁrst term is x 0 ෇ 1 . (With series, we adopt the convention that x 0 ෇ 1 even when x ෇ 0 .) Thus TEC Module 11.2 explores a series that ϱ depends on an angle ␪ in a triangle and enables ͚ x n ෇ 1 ϩ x ϩ x 2 ϩ x 3 ϩ x 4 ϩиии you to see how rapidly the series converges n෇0 when ␪ varies. This is a geometric series with a ෇ 1 and r ෇ x . Since Խ r Խ ෇ Խ x Խ Ͻ 1, it converges and 4 gives ϱ 1 5 ͚ x n ෇ n෇0 1 Ϫ x

ϱ 1 EXAMPLE 7 Show that the series ͚ is convergent, and ﬁnd its sum. n෇1 n͑n ϩ 1͒ SOLUTION This is not a geometric series, so we go back to the deﬁnition of a convergent series and compute the partial sums.

n 1 1 1 1 1 sn ෇ ͚ ෇ ϩ ϩ ϩиииϩ i෇1 i͑i ϩ 1͒ 1 ؒ 2 2 ؒ 3 3 ؒ 4 n͑n ϩ 1͒

We can simplify this expression if we use the partial fraction decomposition

1 ෇ 1 Ϫ 1 i͑i ϩ 1͒ i i ϩ 1

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708 CHAPTER 11 INFINITE SEQUENCES AND SERIES

(see Section 7.4). Thus we have

n n ෇ 1 ෇ ͩ 1 Ϫ 1 ͪ sn ͚ ͚ ෇ i͑i ϩ 1͒ ෇ i i ϩ 1 Notice that the terms cancel in pairs. i 1 i 1 This is an example of a telescoping sum: 1 1 1 1 1 1 1 Because of all the cancellations, the sum ෇ ͩ1 Ϫ ͪ ϩ ͩ Ϫ ͪ ϩ ͩ Ϫ ͪ ϩиииϩͩ Ϫ ͪ collapses (like a pirate’s collapsing 2 2 3 3 4 n n ϩ 1 telescope) into just two terms. ෇ Ϫ 1 1 ϩ Figure 3 illustrates Example 7 by show- n 1 ing the graphs of the sequence of terms ෇ ͞ ͑ ϩ ͒ and the sequence ͕ ͖ 1 an 1 [n n 1 ] sn ෇ ͩ Ϫ ͪ ෇ Ϫ ෇ and so lim sn lim 1 1 0 1 n l ϱ n l ϱ ϩ of partial sums. Notice that an l 0 and n 1

sn l 1. See Exercises76 and 77 for two geometric interpretations of Example 7. Therefore the given series is convergent and

ϱ 1 ͚ ෇ 1 1 n෇1 n͑n ϩ 1͒ ͕ ͖ sn v EXAMPLE 8 Show that the harmonic series

ϱ 1 1 1 1 ͚ ෇ 1 ϩ ϩ ϩ ϩиии n෇1 n 2 3 4 ͕a ͖ n is divergent. 0 n SOLUTION For this particular series it’s convenient to consider the partial sums s2 , s4 , s8, s16, s32, . . . and show that they become large. FIGURE 3 ෇ ϩ 1 s2 1 2 ෇ ϩ 1 ϩ 1 ϩ 1 Ͼ ϩ 1 ϩ 1 ϩ 1 ෇ ϩ 2 s4 1 2 ( 3 4 ) 1 2 ( 4 4 ) 1 2 ෇ ϩ 1 ϩ 1 ϩ 1 ϩ 1 ϩ 1 ϩ 1 ϩ 1 s8 1 2 ( 3 4 ) ( 5 6 7 8 ) Ͼ ϩ 1 ϩ 1 ϩ 1 ϩ 1 ϩ 1 ϩ 1 ϩ 1 1 2 ( 4 4 ) ( 8 8 8 8 ) ෇ ϩ 1 ϩ 1 ϩ 1 ෇ ϩ 3 1 2 2 2 1 2 ෇ ϩ 1 ϩ 1 ϩ 1 ϩ 1 ϩиииϩ 1 ϩ 1 ϩиииϩ 1 s16 1 2 ( 3 4 ) ( 5 8 ) ( 9 16 ) Ͼ ϩ 1 ϩ 1 ϩ 1 ϩ 1 ϩиииϩ 1 ϩ 1 ϩиииϩ 1 1 2 ( 4 4 ) ( 8 8 ) ( 16 16 ) ෇ ϩ 1 ϩ 1 ϩ 1 ϩ 1 ෇ ϩ 4 1 2 2 2 2 1 2

Ͼ ϩ 5 Ͼ ϩ 6 Similarly, s32 1 2 , s64 1 2 , and in general

n s n Ͼ 1 ϩ 2 2 The method used in Example 8 for showing that the harmonic series diverges is due to the This shows that s2n l ϱ as n l ϱ and so ͕sn ͖ is divergent. Therefore the harmonic French scholar Nicole Oresme (1323–1382). series diverges.

6 Theorem If the series ͚ an is convergent, then lim an ෇ 0 . l ϱ n෇1 n

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SECTION 11.2 SERIES 709 ͸ PROOF Let sn ෇ a1 ϩ a2 ϩиииϩan. Then an ෇ sn Ϫ snϪ1. Since an is convergent, the sequence ͕sn ͖ is convergent. Let lim n l ϱ sn ෇ sn. Since Ϫ 1 l ϱ as n l ϱ , we also

have lim n l ϱ snϪ1 ෇ s. Therefore

lim an ෇ lim ͑sn Ϫ snϪ1͒ ෇ lim sn Ϫ lim snϪ1 n l ϱ n l ϱ n l ϱ n l ϱ ෇ s Ϫ s ෇ 0

NOTE 1 With any series ͸ a we associate two sequences: the sequence ͕s ͖ of its par- n ͸ n tial sums and the sequence ͕an ͖ of its terms. If an is convergent, then the limit of the sequence ͕sn ͖ is s (the sum of the series) and, as Theorem 6 asserts, the limit of the sequence ͕an ͖ is 0.

| NOTE 2 The converse of Theorem 6 is not true in general. If lim n l ϱ an ෇ 0, we can- not conclude that ͸ a is convergent. Observe that for the harmonic series ͸ 1͞n we have n ͸ an ෇ 1͞n l 0 as n l ϱ , but we showed in Example 8 that 1͞n is divergent.

7 Test for Divergence If lim an does not exist or if lim an 0 , then the ϱ n l ϱ n l ϱ

series ͚ an is divergent. n෇1

The Test for Divergence follows from Theorem 6 because, if the series is not divergent,

then it is convergent, and so lim n l ϱ an ෇ 0.

ϱ n 2 EXAMPLE 9 Show that the series ͚ 2 diverges. n෇1 5n ϩ 4

SOLUTION n 2 1 1 lim an ෇ lim ෇ lim ෇ 0 n l ϱ n l ϱ 5n 2 ϩ 4 n l ϱ 5 ϩ 4͞n 2 5

So the series diverges by the Test for Divergence. ͸ NOTE 3 If we ﬁnd that lim n l ϱ an 0, we know that an is divergent. If we ﬁnd that lim l ϱ a ෇ 0, we know nothing about the convergence or divergence of ͸ a . Remember n n ͸ n the warning in Note 2: If lim n l ϱ an ෇ 0, the series an might converge or it might diverge.

8 Theorem If ͸ a and ͸ b are convergent series, then so are the series ͸ ca n ͸ n ͸ n (where c is a constant), ͑an ϩ bn ͒ , and ͑an Ϫ bn ͒ , and ϱ ϱ ϱ ϱ ϱ

(i) ͚ can ෇ c ͚ an (ii) ͚ ͑an ϩ bn ͒ ෇ ͚ an ϩ ͚ bn n෇1 n෇1 n෇1 n෇1 n෇1

ϱ ϱ ϱ

(iii) ͚ ͑an Ϫ bn ͒ ෇ ͚ an Ϫ ͚ bn n෇1 n෇1 n෇1

These properties of convergent series follow from the corresponding Limit Laws for Sequences in Section 11.1. For instance, here is how part (ii) of Theorem 8 is proved: Let n ϱ n ϱ

sn ෇ ͚ ai s ෇ ͚ an tn ෇ ͚ bi t ෇ ͚ bn i෇1 n෇1 i෇1 n෇1

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710 CHAPTER 11 INFINITE SEQUENCES AND SERIES ͸ The nth partial sum for the series ͑an ϩ bn ͒ is

un ෇ ͚ ͑ai ϩ bi ͒ i෇1 and, using Equation 5.2.10, we have

n n n ෇ ͑ ϩ ͒ ෇ ͚ͩ ϩ ͚ ͪ lim un lim ͚ ai bi lim ai bi l ϱ l ϱ l ϱ n n i෇1 n i෇1 i෇1

n n

෇ lim ͚ ai ϩ lim ͚ bi l ϱ l ϱ n i෇1 n i෇1

෇ lim sn ϩ lim tn ෇ s ϩ t n l ϱ n l ϱ ͸ Therefore ͑an ϩ bn ͒ is convergent and its sum is

ϱ ϱ ϱ

͚ ͑an ϩ bn ͒ ෇ s ϩ t ෇ ͚ an ϩ ͚ bn n෇1 n෇1 n෇1

ϱ 3 1 EXAMPLE 10 ͩ ϩ ͪ Find the sum of the series ͚ n . n෇1 n͑n ϩ 1͒ 2 ͸ ͞ n ෇ 1 ෇ 1 SOLUTION The series 1 2 is a geometric series with a 2 and r 2 , so ϱ 1 1 ͚ ෇ 2 ෇ 1 n Ϫ 1 n෇1 2 1 2 In Example 7 we found that ϱ 1 ͚ ෇ 1 n෇1 n͑n ϩ 1͒ So, by Theorem 8, the given series is convergent and

ϱ ϱ ϱ ͩ 3 ϩ 1 ͪ ෇ ͚ 1 ϩ ͚ 1 ͚ n 3 n n෇1 n͑n ϩ 1͒ 2 n෇1 n͑n ϩ 1͒ n෇1 2 ෇ 3 ؒ 1 ϩ 1 ෇ 4

NOTE 4 A ﬁnite number of terms doesn’t affect the convergence or divergence of a series. For instance, suppose that we were able to show that the series ϱ n ͚ 3 n෇4 n ϩ 1 is convergent. Since ϱ ϱ n ෇ 1 ϩ 2 ϩ 3 ϩ n ͚ 3 ͚ 3 n෇1 n ϩ 1 2 9 28 n෇4 n ϩ 1

͸ϱ 3 it follows that the entire series n෇1 n͑͞n ϩ 1͒ is convergent. Similarly, if it is known that ͸ϱ the series n෇Nϩ1 an converges, then the full series

ϱ N ϱ

͚ an ෇ ͚ an ϩ ͚ an n෇1 n෇1 n෇Nϩ1 is also convergent.

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SECTION 11.2 SERIES 711

11.2 Exercises

1. (a) What is the difference between a sequence and a series? ϱ ϱ 10 n 21. ͑ ͒nϪ1 22. ͚ 6 0.9 ͚ nϪ1 (b) What is a convergent series? What is a divergent series? n෇1 n෇1 ͑Ϫ9͒ ϱ 2. ͸ ෇ ϱ Ϫ ϱ Explain what it means to say that n෇1 an 5 . ͑Ϫ3͒n 1 1 23. 24. ͚ n ͚ n ͸ϱ ෇ ෇ s 3–4 Calculate the sum of the series n෇1 an whose partial sums n 1 4 n 0 ( 2 ) are given. ϱ ␲ n ϱ e n 25. 26. 2 ͚ nϩ1 ͚ nϪ1 n Ϫ 1 ෇ 3 ෇ 3 3. s ෇ 2 Ϫ 3͑0.8͒n 4. s ෇ n 0 n 1 n n 4n 2 ϩ 1 27–42 Determine whether the series is convergent or divergent. 5–8 Calculate the first eight terms of the sequence of partial If it is convergent, find its sum. sums correct to four decimal places. Does it appear that the series 1 1 1 1 1 27. ϩ ϩ ϩ ϩ ϩиии is convergent or divergent? 3 6 9 12 15 ϱ 1 ϱ 1 5. ͚ 6. ͚ 1 2 1 2 1 2 3 ͑ ϩ ͒ 28. ϩ ϩ ϩ ϩ ϩ ϩиии n෇1 n n෇1 ln n 1 3 9 27 81 243 729 ϱ ϱ Ϫ ͑Ϫ ͒n 1 ϱ ϱ n 1 n Ϫ 1 k͑k ϩ 2͒ 7. ͚ 8. ͚ 29. ͚ 30. ͚ n෇1 1 ϩ sn n෇1 n! 2 n෇1 3n Ϫ 1 k෇1 ͑k ϩ 3͒ ϱ 1 ϩ 2n ϱ 1 ϩ 3 n 31. ͚ 32. ͚ ; 9–14 Find at least 10 partial sums of the series. Graph both the n n n෇1 3 n෇1 2 sequence of terms and the sequence of partial sums on the same ϱ ϱ screen. Does it appear that the series is convergent or divergent? Ϫ 33. ͚ sn 2 34. ͚ ͓͑0.8͒n 1 Ϫ ͑0.3͒n͔ If it is convergent, find the sum. If it is divergent, explain why. n෇1 n෇1

ϱ ϱ 12 ϱ 2 ϱ 9. 10. n ϩ 1 1 ͚ n ͚ cos n 35. ͩ ͪ 36. ͚ ͑Ϫ ͒ ͚ ln n n෇1 5 n෇1 2 ϩ ϩ 2 n෇1 2n 1 n෇1 1 (3) ϱ n ϱ 7 nϩ1 11. 12. ϱ k ϱ ͚ ͚ n ␲ n෇1 sn 2 ϩ 4 n෇1 10 37. ͚ ͩ ͪ 38. ͚ ͑cos 1͒k k෇0 3 k෇1 ϱ 1 1 ϱ 1 13. ͚ ͩ Ϫ ͪ 14. ϱ ϱ ͚ ͑ ϩ 3 2 n෇1 sn sn ϩ 1 n෇2 n n 2͒ 39. 40. ͩ ϩ ͪ ͚ arctan n ͚ n n෇1 n෇1 5 n

ϱ 1 1 ϱ en 2n 41. ͚ ͩ ϩ ͪ 42. ͚ 15. Let an ෇ . n ͑ ϩ ͒ 2 3n ϩ 1 n෇1 e n n 1 n෇1 n

(a) Determine whether ͕an ͖ is convergent. ϱ (b) Determine whether ͸ ෇ a is convergent. n 1 n 43–48 Determine whether the series is convergent or divergent 16. (a) Explain the difference between by expressing sn as a telescoping sum (as in Ex am ple 7). If it is n n convergent, ﬁnd its sum. ͚ ai and ͚ aj i෇1 j෇1 ϱ 2 ϱ n 43. ͚ 44. ͚ ln (b) Explain the difference between 2 n෇2 n Ϫ 1 n෇1 n ϩ 1 n n ϱ 3 ͚ ai and ͚ aj 45. i෇1 i෇1 ͚ n෇1 n͑n ϩ 3͒

17–26 Determine whether the geometric series is convergent or ϱ 1 1 46. ͩ Ϫ ͪ divergent. If it is convergent, ﬁnd its sum. ͚ cos 2 cos 2 n෇1 n ͑n ϩ 1͒ 17. Ϫ ϩ 16 Ϫ 64 ϩиии 18. ϩ ϩ 9 ϩ 27 ϩ иии 3 4 3 9 4 3 4 16 ϱ ϱ 1 47. 1͞n Ϫ 1͑͞nϩ1͒ 48. 19. 10 Ϫ 2 ϩ 0.4 Ϫ 0.08 ϩиии ͚ (e e ) ͚ 3 n෇1 n෇2 n Ϫ n 20. 2 ϩ 0.5 ϩ 0.125 ϩ 0.03125 ϩиии

; Graphing calculator or computer required CAS Computer algebra system required 1. Homework Hints available at stewartcalculus.com

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712 CHAPTER 11 INFINITE SEQUENCES AND SERIES

49. Let x ෇ 0.99999 . . . . ͸ϱ Ϫn 68. If the nth partial sum of a series n෇1 an is sn ෇ 3 Ϫ n2 , x Ͻ x ෇ (a) Do you think that 1 or 1 ? ͸ϱ (b) Sum a geometric series to find the value of x . find an and n෇1 an . (c) How many decimal representations does the number 1 69. A patient takes 150 mg of a drug at the same time every day. have? Just before each tablet is taken, 5% of the drug remains in the (d) Which numbers have more than one decimal body. representation? (a) What quantity of the drug is in the body after the third 50. A sequence of terms is defined by tablet? After the n th tablet? (b) What quantity of the drug remains in the body in the long ෇ ෇ ͑ Ϫ ͒ a1 1 an 5 n anϪ1 run? ͸ϱ Calculate n෇1 an. 70. After injection of a dose D of insulin, the concentration of insulin in a patient’s system decays exponentially and so it 51–56 Express the number as a ratio of integers. can be written as DeϪat , where t represents time in hours and 51. 0.8 ෇ 0.8888 . . . 52. 0.46 ෇ 0.46464646 . . . a is a positive constant. (a) If a dose DT is injected every hours, write an expression 53. ෇ 2.516 2.516516516 . . . for the sum of the residual concentrations just before the 54. 10.135 ෇ 10.135353535 . . . ͑n ϩ 1͒st injection. (b) Determine the limiting pre-injection concentration. 55. 56. 1.5342 7.12345 (c) If the concentration of insulin must always remain at or above a critical value CD , determine a minimal dosage in terms of Ca , , and T . 57–63 Find the values of x for which the series converges. Find the sum of the series for those values of x . 71. When money is spent on goods and services, those who ϱ ϱ receive the money also spend some of it. The people receiv- 57. ͚ ͑Ϫ5͒nx n 58. ͚ ͑x ϩ 2͒n ing some of the twice-spent money will spend some of that, n෇1 n෇1 and so on. Economists call this chain reaction the multiplier ϱ ͑x Ϫ 2͒n ϱ effect. In a hypothetical isolated community, the local 59. 60. ͑Ϫ ͒n͑ Ϫ ͒n ͚ n ͚ 4 x 5 n෇0 3 n෇0 government begins the process by spending D dollars. Sup- pose that each recipient of spent money spends 100c% and ϱ 2n ϱ sin n x 61. 62. saves 100s% of the money that he or she receives. The val- ͚ n ͚ n n෇0 x n෇0 3 ues c and s are called the marginal propensity to consume ϱ and the marginal propensity to save and, of course, nx 63. ͚ e c ϩ s ෇ 1. n෇0 (a) Let Sn be the total spending that has been generated after

nStransactions. Find an equation for n . ෇ ෇ ͞ 64. We have seen that the harmonic series is a divergent series (b) Show that limn l ϱ Sn kD, where k 1 s . The number whose terms approach 0. Show that k is called the multiplier. What is the multiplier if the marginal propensity to consume is 80% ? ϱ 1 Note: The federal government uses this principle to justify ͚ lnͩ1 ϩ ͪ deficit spending. Banks use this principle to justify lend ing a n෇1 n large percentage of the money that they receive in deposits. is another series with this property. 72. A certain ball has the property that each time it falls from a height h onto a hard, level surface, it rebounds to a height CAS 65–66 Use the partial fraction command on your CAS to find rh, where 0 Ͻ r Ͻ 1 . Suppose that the ball is dropped from a convenient expression for the partial sum, and then use this an initial height of H meters. expression to find the sum of the series. Check your answer by (a) Assuming that the ball continues to bounce indefinitely, using the CAS to sum the series directly. find the total distance that it travels. ϱ ϱ (b) Calculate the total time that the ball travels. (Use the 3n2 ϩ 3n ϩ 1 1 65. 66. 1 t 2 ͚ 2 3 ͚ 5 3 fact that the ball falls 2 t meters in t seconds .) ෇ ͑n ϩ n͒ ෇ n Ϫ n ϩ n n 1 n 3 5 4 (c) Suppose that each time the ball strikes the surface with velocity v it rebounds with velocity Ϫkv , where 0 Ͻ k Ͻ 1. How long will it take for the ball to come 67. ͸ϱ If the nth partial sum of a series n෇1 an is to rest? n Ϫ 1 73. Find the value of c if sn ෇ n ϩ 1 ϱ ͚ ͑ ϩ c͒Ϫn ෇ ͸ϱ 1 2 find an and n෇1 an . n෇2

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CHAPTER 11.2 SERIES 713

74. Find the value of c such that 79. What is wrong with the following calculation? ϱ 0 ෇ 0 ϩ 0 ϩ 0 ϩиии ͚ e nc ෇ 10 ෇ n 0 ෇ ͑1 Ϫ 1͒ ϩ ͑1 Ϫ 1͒ ϩ ͑1 Ϫ 1͒ ϩиии 75. In Example 8 we showed that the harmonic series is diver- ෇ 1 Ϫ 1 ϩ 1 Ϫ 1 ϩ 1 Ϫ 1 ϩиии gent. Here we outline another method, making use of the fact that e x Ͼ 1 ϩ xx for any Ͼ 0 . (See Exercise 4.3.78.) ෇ 1 ϩ ͑Ϫ1 ϩ 1͒ ϩ ͑Ϫ1 ϩ 1͒ ϩ ͑Ϫ1 ϩ 1͒ ϩиии If sn is the nth partial sum of the harmonic series, show that s ෇ 1 ϩ 0 ϩ 0 ϩ 0 ϩиии෇ 1 e n Ͼ n ϩ 1 . Why does this imply that the harmonic series is divergent? (Guido Ubaldus thought that this proved the existence of God ; 76. Graph the curves y ෇ x n , 0 ഛ x ഛ 1 , for n ෇ 0, 1, 2, 3, 4, . . . because “something has been created out of nothing.”) ͸ϱ on a common screen. By finding the areas between successive 80. Suppose that n෇1 an ͑an 0͒ is known to be a convergent ͸ϱ curves, give a geometric demonstration of the fact, shown in series. Prove that n෇1 1͞an is a divergent series. Example 7, that 81. Prove part (i) of Theorem 8. ϱ 1 ෇ ͸ ͸ ͚ 1 82. If an is divergent and c 0 , show that can is divergent. n෇1 n͑n ϩ 1͒ ͸ ͸ 83. If an is convergent and bn is divergent, show that 77. The figure shows two circles and DC of radius 1 that touch ͸ the series ͑an ϩ bn͒ is divergent. [Hint: Argue by at . TP is a common tangent line; C1 is the circle that touches contradiction.] , , and TDC ; C2 is the circle that touches , DC , and C1 ; C3 is 84. ͸ ͸ ͸ ͑ ϩ ͒ the circle that touches , DC , and C2 . This procedure can be If an and bn are both divergent, is an bn neces- continued indefinitely and produces an infinite sequence of sarily divergent? circles ͕C ͖ . Find an expression for the diameter of C and n n 85. Suppose that a series ͸ a has positive terms and its partial thus provide another geometric demonstration of Example 7. n sums s satisfy the inequality s ഛ 1000 for all n . Explain why ͸ n n an must be convergent. 86. The Fibonacci sequence was defined in Section 11.1 by the equations

f1 ෇ 1, f2 ෇ 1, fn ෇ fnϪ1 ϩ fnϪ2 n ജ 3 P Show that each of the following statements is true. C£ C™ 1 1 1 11(a) ෇ Ϫ fnϪ1 fnϩ1 fnϪ1 fn fn fnϩ1

C D ϱ C¡ 1 (b) ͚ ෇ 1 T n෇2 fnϪ1 fnϩ1 ϱ fn 78. A right triangle ABC is given with ЄA ෇ ␪ and Խ AC Խ ෇ b . (c) ͚ ෇ 2 ෇ f Ϫ f ϩ CD is drawn perpendicular to , DEAB is drawn perpendicular n 2 n 1 n 1 to BC , EF Ќ AB , and this process is continued indefi nitely, 87. The Cantor set, named after the German mathematician Georg as shown in the figure. Find the total length of all the Cantor (1845–1918), is constructed as follows. We start with perpendiculars 1 2 the closed interval [0, 1] and remove the open interval ( 3 , 3 ) . 1 2 That leaves the two intervals [0, 3 ] and [ 3, 1] and we remove Խ CD Խ ϩ Խ DE Խ ϩ Խ EF Խ ϩ Խ FG Խ ϩиии the open middle third of each. Four intervals remain and again we remove the open middle third of each of them. We continue this procedure indefinitely, at each step removing the open in terms of b and ␪ . middle third of every interval that remains from the preceding A step. The Cantor set consists of the numbers that remain in [0, 1] after all those intervals have been removed. ¨ D (a) Show that the total length of all the intervals that are F removed is 1. Despite that, the Cantor set contains infi- H b nitely many numbers. Give examples of some numbers in the Cantor set. (b) The Sierpinski carpet is a two-dimensional counterpart of the Cantor set. It is constructed by removing the center B CEG one-ninth of a square of side 1, then removing the centers

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714 CHAPTER 11 INFINITE SEQUENCES AND SERIES

of the eight smaller remaining squares, and so on. (The fig- (b) Use mathematical induction to prove your guess. ure shows the first three steps of the construction.) Show (c) Show that the given infinite series is convergent, and find that the sum of the areas of the removed squares is 1. This its sum. implies that the Sierpinski carpet has area 0. 90. In the figure there are infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of length 1, find the total area occupied by the circles.

88. (a) A sequence ͕an ͖ is deﬁned recursively by the equation ෇ 1 ͑ ϩ ͒ ജ an 2 anϪ1 anϪ2 for n 3 , where a1 and a2 can be any real numbers. Experiment with various values of a1 and a2 and use your calculator to guess the limit of the sequence.

(b) Find limn l ϱ an in terms of a1 and a2 by expressing

anϩ1 Ϫ an in terms of a2 Ϫ a1 and summing a series. ͸ϱ 89. Consider the series n෇1 n͑͞n ϩ 1͒!.

(a) Find the partial sums s1, s2, s3, and s4 . Do you recognize the

denominators? Use the pattern to guess a formula for sn .

11.3 The Integral Test and Estimates of Sums

In general, it is difﬁcult to ﬁnd the exact sum of a series. We were able to accomplish this for geometric series and the series ͸ 1͓͞n͑n ϩ 1͔͒ because in each of those cases we could

find a simple formula for the nth partial sum sn . But usually it isn’t easy to discover such a formula. Therefore, in the next few sections, we develop several tests that enable us to determine whether a series is convergent or divergent without explicitly finding its sum. (In some cases, however, our methods will enable us to find good estimates of the sum.) Our first test involves improper integrals. We begin by investigating the series whose terms are the reciprocals of the squares of the positive integers: n 1 ϱ n s ෇ ͚ 1 1 1 1 1 1 n 2 ෇ ϩ ϩ ϩ ϩ ϩиии i෇1 i ͚ 2 2 2 2 2 2 n෇1 n 1 2 3 4 5 5 1.4636 There’s no simple formula for the sum s of the first n terms, but the computer-generated 10 1.5498 n table of approximate values given in the margin suggests that the partial sums are approach- 50 1.6251 l ϱ 100 1.6350 ing a number near 1.64 as n and so it looks as if the series is convergent. 500 1.6429 We can confirm this impression with a geometric argument. Figure 1 shows the curve ෇ ͞ 2 1000 1.6439 y 1 x and rectangles that lie below the curve. The base of each rectangle is an interval ෇ ͞ 2 5000 1.6447 of length 1; the height is equal to the value of the function y 1 x at the right endpoint of the interval. y 1 y= ≈

area= 1 1@

0 1 2 3 4 5 x area= 1 area= 1 area= 1 area= 1 FIGURE 1 2@ 3@ 4@ 5@

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SECTION 11.3 THE INTEGRAL TEST AND ESTIMATES OF SUMS 715

So the sum of the areas of the rectangles is

ϱ 1 ϩ 1 ϩ 1 ϩ 1 ϩ 1 ϩиии෇ 1 2 2 2 2 2 ͚ 2 1 2 3 4 5 n෇1 n

If we exclude the ﬁrst rectangle, the total area of the remaining rectangles is smaller than ෇ ͞ 2 ജ xϱ ͑ ͞ 2 ͒ the area under the curve y 1 x for x 1 , which is the value of the integral 1 1 x dx . In Section 7.8 we discovered that this improper integral is convergent and has value 1. So the picture shows that all the partial sums are less than

1 ϩ ϱ 1 ෇ 2 y 2 dx 2 1 1 x

Thus the partial sums are bounded. We also know that the partial sums are increasing (because all the terms are positive). Therefore the partial sums converge (by the Monotonic Sequence Theorem) and so the series is convergent. The sum of the series (the limit of the partial sums) is also less than 2:

ϱ 1 1 1 1 1 ෇ ϩ ϩ ϩ ϩиииϽ ͚ 2 2 2 2 2 2 n෇1 n 1 2 3 4

[The exact sum of this series was found by the Swiss mathematician Leonhard Euler (1707–1783) to be ␲ 2͞6 , but the proof of this fact is quite difﬁcult. (See Problem 6 in the Problems Plus following Chapter 15.)] Now let’s look at the series

ϱ 1 ෇ 1 ϩ 1 ϩ 1 ϩ 1 ϩ 1 ϩиии n 1 ͚ ෇ s s s s s s n sn ෇ ͚ n 1 n 1 2 3 4 5 i෇1 si

5 3.2317 The table of values of sn suggests that the partial sums aren’t approaching a ﬁnite number, 10 5.0210 so we suspect that the given series may be divergent. Again we use a picture for conﬁrma- 50 12.7524 tion. Figure 2 shows the curve y ෇ 1͞sx , but this time we use rectangles whose tops lie 100 18.5896 above the curve. 500 43.2834 1000 61.8010 y y= 1 5000 139.9681 œ„x

0 1 2 3 4 5 x area= 1 area= 1 area= 1 area= 1 FIGURE 2 œ„1 œ2„ œ3„ œ4„

The base of each rectangle is an interval of length 1. The height is equal to the value of the function y ෇ 1͞sx at the left endpoint of the interval. So the sum of the areas of all the rectangles is

ϱ 1 ϩ 1 ϩ 1 ϩ 1 ϩ 1 ϩиии෇ ͚ 1 s1 s2 s3 s4 s5 n෇1 sn

This total area is greater than the area under the curve y ෇ 1͞sx for x ജ 1 , which is equal

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716 CHAPTER 11 INFINITE SEQUENCES AND SERIES

xϱ ͞s to the integral 1 (1 x ) dx. But we know from Section 7.8 that this improper integral is divergent. In other words, the area under the curve is inﬁnite. So the sum of the series must be inﬁnite; that is, the series is divergent. The same sort of geometric reasoning that we used for these two series can be used to prove the following test. (The proof is given at the end of this section.)

The Integral Test Suppose f is a continuous, positive, decreasing function on ͓1, ϱ͒ ͸ϱ and let an ෇ f ͑n͒ . Then the series n෇1 an is convergent if and only if the improper xϱ ͑ ͒ integral 1 f x dx is convergent. In other words: ϱ ϱ (i) If y f ͑x͒ dx is convergent, then ͚ an is convergent. 1 n෇1 ϱ ϱ (ii) If y f ͑x͒ dx is divergent, then ͚ an is divergent. 1 n෇1

NOTE When we use the Integral Test, it is not necessary to start the series or the integral at n ෇ 1 . For instance, in testing the series

ϱ 1 ϱ 1 ͚ 2 we use y 2 dx n෇4 ͑n Ϫ 3͒ 4 ͑x Ϫ 3͒

Also, it is not necessary that f be always decreasing. What is important is that f be ulti- ͸ϱ mately decreasing, that is, decreasing for x larger than some number N . Then n෇N an is ͸ϱ convergent, so n෇1 an is convergent by Note 4 of Section 11.2.

ϱ 1 EXAMPLE 1 Test the series ͚ 2 for convergence or divergence. n෇1 n ϩ 1

SOLUTION The function f ͑x͒ ෇ 1͑͞x 2 ϩ 1͒ is continuous, positive, and decreasing on ͓1, ϱ͒ so we use the Integral Test:

ϱ 1 t 1 t ෇ ෇ Ϫ1 y 2 dx lim y 2 dx lim tan x]1 1 x ϩ 1 t l ϱ 1 x ϩ 1 t l ϱ ␲ ␲ ␲ ␲ ෇ lim ͩtanϪ1t Ϫ ͪ ෇ Ϫ ෇ t l ϱ 4 2 4 4

xϱ ͑͞ 2 ϩ ͒ Thus 1 1 x 1 dx is a convergent integral and so, by the Integral Test, the series ͸ 1͑͞n2 ϩ 1͒ is convergent.

ϱ 1 v EXAMPLE 2 For what values of p is the series ͚ p convergent? n෇1 n

p p SOLUTION If p Ͻ 0 , then limn l ϱ ͑1͞n ͒ ෇ ϱ. If p ෇ 0 , then limn l ϱ ͑1͞n ͒ ෇ 1. In p either case limn l ϱ ͑1͞n ͒ 0, so the given series diverges by the Test for Divergence (11.2.7). In order to use the Integral Test we need to be If p Ͼ 0 , then the function f ͑x͒ ෇ 1͞x p is clearly continuous, positive, and decreasing able to evaluate xϱ ͑ ͒ and therefore we 1 f x dx on ͓1, ϱ͒ . We found in Chapter 7 [see (7.8.2)] that have to be able to ﬁnd an antiderivative of f . Frequently this is difﬁcult or impossible, so we ϱ 1 need other tests for convergence too. Ͼ ഛ y p dx converges if p 1 and diverges if p 1 1 x

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SECTION 11.3 THE INTEGRAL TEST AND ESTIMATES OF SUMS 717

It follows from the Integral Test that the series ͸ 1͞n p converges if p Ͼ 1 and diverges if 0 Ͻ p ഛ 1 . (For p ෇ 1 , this series is the harmonic series discussed in Example 8 in Section 11.2.)

The series in Example 2 is called the p-series. It is important in the rest of this chapter, so we summarize the results of Example 2 for future reference as follows.

ϱ 1 1 Ͼ ഛ The p -series ͚ p is convergent if p 1 and divergent if p 1 . n෇1 n

EXAMPLE 3 (a) The series ϱ 1 ෇ 1 ϩ 1 ϩ 1 ϩ 1 ϩиии ͚ 3 3 3 3 3 n෇1 n 1 2 3 4 is convergent because it is a p-series with p ෇ 3 Ͼ 1 . (b) The series ϱ ϱ ͚ 1 ෇ ͚ 1 ෇ ϩ 1 ϩ 1 ϩ 1 ϩиии 1͞3 3 1 3 3 3 n෇1 n n෇1 sn s2 s3 s4 ෇ 1 Ͻ is divergent because it is a p-series with p 3 1 .

NOTE We should not infer from the Integral Test that the sum of the series is equal to the value of the integral. In fact,

ϱ ␲ 2 ϱ 1 ෇ 1 ෇ ͚ 2 whereas y 2 dx 1 n෇1 n 6 1 x

Therefore, in general, ϱ ϱ ͚ an y f ͑x͒ dx n෇1 1 ϱ ln n v EXAMPLE 4 Determine whether the series ͚ converges or diverges. n෇1 n SOLUTION The function f ͑x͒ ෇ ͑ln x͒͞xxis positive and continuous for Ͼ 1 because the logarithm function is continuous. But it is not obvious whether or not f is decreasing, so we compute its derivative: ͑1͞x͒x Ϫ ln x 1 Ϫ ln x f Ј͑x͒ ෇ ෇ x 2 x 2

Thus f Ј͑x͒ Ͻ 0 when ln x Ͼ 1 , that is, x Ͼ ef . It follows that is decreasing when xϾ e and so we can apply the Integral Test:

t 2 ϱ ln x t ln x ͑ln x͒ y dx ෇ lim y dx ෇ lim ͬ 1 t l ϱ 1 t l ϱ x x 2 1 ͑ln t͒2 ෇ lim ෇ ϱ t l ϱ 2 Since this improper integral is divergent, the series ͸ ͑ln n͒͞n is also divergent by the Integral Test.

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718 CHAPTER 11 INFINITE SEQUENCES AND SERIES

Estimating the Sum of a Series ͸ Suppose we have been able to use the Integral Test to show that a series an is convergent and we now want to ﬁnd an approximation to the sum s of the series. Of course, any partial

sum sn is an approximation to s because limn l ϱ sn ෇ s. But how good is such an approximation? To ﬁnd out, we need to estimate the size of the remainder

Rn ෇ s Ϫ sn ෇ anϩ1 ϩ anϩ2 ϩ anϩ3 ϩиии

y The remainder Rn is the error made when sn , the sum of the ﬁrst n terms, is used as an approx- y=ƒ imation to the total sum. We use the same notation and ideas as in the Integral Test, assuming that f is decreasing on ͓n, ϱ͒ . Comparing the areas of the rectangles with the area under y ෇ f ͑x͒ for x Ͼ n in Figure 3, we see that

ϱ an+1 a n+2 . . . Rn ෇ anϩ1 ϩ anϩ2 ϩиииഛy f ͑x͒ dx n 0 n x Similarly, we see from Figure 4 that

FIGURE 3 ϱ Rn ෇ anϩ1 ϩ anϩ2 ϩиииജy f ͑x͒ dx nϩ1 y y=ƒ So we have proved the following error estimate.

2 Remainder Estimate for the Integral Test Suppose f ͑k͒ ෇ a , where f is a ͸ k continuous, positive, decreasing function for x ജ n and an is convergent. If an+1 an+2 . . . Rn ෇ s Ϫ sn, then x 0 n+1 ϱ ϱ y f ͑x͒ dx ഛ Rn ഛ y f ͑x͒ dx nϩ1 n FIGURE 4

v EXAMPLE 5 (a) Approximate the sum of the series ͸ 1͞n 3 by using the sum of the ﬁrst 10 terms. Estimate the error involved in this approximation. (b) How many terms are required to ensure that the sum is accurate to within 0.0005 ? xϱ ͑ ͒ ͑ ͒ ෇ ͞ 3 SOLUTION In both parts (a) and (b) we need to know n f x dx . With f x 1 x , which satisﬁes the conditions of the Integral Test, we have

t ϱ 1 1 1 1 1 y dx ෇ lim ͫϪ ͬ ෇ lim ͩϪ ϩ ͪ ෇ n 3 t l ϱ 2 t l ϱ 2 2 2 x 2x n 2t 2n 2n (a) Approximating the sum of the series by the 10th partial sum, we have

ϱ 1 Ϸ ෇ 1 ϩ 1 ϩ 1 ϩиииϩ 1 Ϸ ͚ 3 s10 3 3 3 3 1.1975 n෇1 n 1 2 3 10

According to the remainder estimate in 2 , we have

ഛ ϱ 1 ෇ 1 ෇ 1 R10 y 3 dx 2 10 x 2͑10͒ 200

So the size of the error is at most 0.005 .

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SECTION 11.3 THE INTEGRAL TEST AND ESTIMATES OF SUMS 719

(b) Accuracy to within 0.0005 means that we have to ﬁnd a value of n such that

Rn ഛ 0.0005. Since ϱ 1 1 Rn ഛ y dx ෇ n x 3 2n2 1 we want Ͻ 0.0005 2n2 Solving this inequality, we get 1 n2 Ͼ ෇ 1000 or n Ͼ s1000 Ϸ 31.6 0.001 We need 32 terms to ensure accuracy to within 0.0005 .

If we add sn to each side of the inequalities in 2 , we get

ϱ ϱ 3 sn ϩ y f ͑x͒ dx ഛ s ഛ sn ϩ y f ͑x͒ dx nϩ1 n

because sn ϩ Rn ෇ s . The inequalities in 3 give a lower bound and an upper bound for s . They provide a more accurate approximation to the sum of the series than the partial sum

sn does. ϱ 1 Although Euler was able to calculate the exact EXAMPLE 6 ෇ Use 3 with n 10 to estimate the sum of the series ͚ 3 . sum of the pp -series for ෇ 2 , nobody has been n෇1 n able to ﬁnd the exact sum for p ෇ 3 . In Example SOLUTION The inequalities in 3 become 6, however, we show how to estimate this sum. ϩ ϱ 1 ഛ ഛ ϩ ϱ 1 s10 y 3 dx s s10 y 3 dx 11 x 10 x From Example 5 we know that

ϱ 1 1 y dx ෇ n x 3 2n 2 1 1 so s ϩ ഛ s ഛ s ϩ 10 2͑11͒2 10 2͑10͒2

Using s10 Ϸ 1.197532, we get

1.201664 ഛ s ഛ 1.202532

If we approximate s by the midpoint of this interval, then the error is at most half the length of the interval. So

ϱ 1 Ϸ Ͻ ͚ 3 1.2021 with error 0.0005 n෇1 n

If we compare Example 6 with Example 5, we see that the improved estimate in 3 can

be much better than the estimate s Ϸ sn . To make the error smaller than 0.0005 we had to use 32 terms in Example 5 but only 10 terms in Example 6.

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720 CHAPTER 11 INFINITE SEQUENCES AND SERIES

Proof of the Integral Test

y y=ƒ We have already seen the basic idea behind the proof of the Integral Test in Figures 1 and ͸ 2 ͸ ͸ 2 for the series 1͞n and 1͞sn . For the general series an , look at Figures 5 and 6. The area of the ﬁrst shaded rectangle in Figure 5 is the value off at the right endpoint of͓1, 2͔ ,

that is, f ͑2͒ ෇ a2 . So, comparing the areas of the shaded rectangles with the area under y ෇ f ͑x͒ from 1 to n , we see that

a™ a£ a¢ a∞ an . . . n 0 12345 n x 4 a2 ϩ a3 ϩиииϩan ഛ y f ͑x͒ dx 1 FIGURE 5 (Notice that this inequality depends on the fact that f is decreasing.) Likewise, Figure 6 shows that y y=ƒ n 5 y f ͑x͒ dx ഛ a1 ϩ a2 ϩиииϩanϪ1 1

an-1 ϱ (i) If y f ͑x͒ dx is convergent, then 4 gives a¡ a™ a£ a¢ 1 n 0 12345. . . n x n ϱ ͚ ai ഛ y f ͑x͒ dx ഛ y f ͑x͒ dx i෇2 1 1 FIGURE 6 since f ͑x͒ ജ 0 . Therefore n ϱ sn ෇ a1 ϩ ͚ ai ഛ a1 ϩ y f ͑x͒ dx ෇ M, say i෇2 1

Since sn ഛ Mn for all , the sequence ͕sn ͖ is bounded above. Also

snϩ1 ෇ sn ϩ anϩ1 ജ sn

since a ϩ ෇ f ͑n ϩ 1͒ ജ 0. Thus ͕s ͖ is an increasing bounded sequence and so it is con- n 1 n ͸ vergent by the Monotonic Sequence Theorem (11.1.12). This means that an is convergent. xϱ ͑ ͒ xn ͑ ͒ l ϱ l ϱ ͑ ͒ ജ (ii) If 1 f x dx is divergent, then 1 f x dx as n because f x 0 . But 5 gives nϪ1 n y f ͑x͒ dx ഛ ͚ ai ෇ snϪ1 1 i෇1 ͸ and so snϪ1 l ϱ . This implies that sn l ϱ and so an diverges.

11.3 Exercises

1. Draw a picture to show that 3–8 Use the Integral Test to determine whether the series is ϱ convergent or divergent. 1 ϱ 1 Ͻ ͚ 1.3 y 1.3 dx ϱ ϱ ෇ 1 1 1 n 2 n x 3. 4. ͚ ͚ 5 ෇ s5 ෇ n What can you conclude about the series? n 1 n n 1 ϱ 1 ϱ 1 2. Suppose f is a continuous positive decreasing function for 5. 6. ͚ 3 ͚ n෇1 ͑2n ϩ 1͒ n෇1 s ϩ x ജ 1 and an ෇ f ͑n͒ . By drawing a picture, rank the following n 4 ϱ ϱ three quantities in increasing order: n 3 7. 8. 2 Ϫn ͚ 2 ͚ n e 5 6 ෇ ϩ ෇ 6 n 1 n 1 n 1 y f ͑x͒ dx ͚ ai ͚ ai 1 i෇1 i෇2

CAS Computer algebra system required 1. Homework Hints available at stewartcalculus.com

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CHAPTER 11.3 THE INTEGRAL TEST AND ESTIMATES OF SUMS 721

9–26 Determine whether the series is convergent or divergent. 34. Leonhard Euler was able to calculate the exact sum of the ϱ ϱ pp-series with ෇ 2 : 1 Ϫ0.9999 9. ͚ 10. ͚ n ϱ s2 ␲ 2 n෇1 n n෇3 ␨͑ ͒ ෇ 1 ෇ 2 ͚ 2 n෇1 n 6 1 1 1 1 11. 1 ϩ ϩ ϩ ϩ ϩиии 8 27 64 125 (See page 715.) Use this fact to find the sum of each series. ϱ 1 ϱ 1 1 1 1 1 (a) ͚ 2 (b) ͚ 2 12. 1 ϩ ϩ ϩ ϩ ϩиии n෇2 n n෇3 ͑n ϩ 1͒ 2s2 3s3 4s4 5s5 ϱ 1 (c) ͚ 2 1 1 1 1 n෇1 ͑2n͒ 13. 1 ϩ ϩ ϩ ϩ ϩиии 3 5 7 9 35. Euler also found the sum of the pp -series with ෇ 4 : 1 1 1 1 1 14. ϩ ϩ ϩ ϩ ϩиии ϱ ␲ 4 5 8 11 14 17 ␨͑ ͒ ෇ 1 ෇ 4 ͚ 4 n෇1 n 90 ϱ sn ϩ 4 ϱ n2 15. 16. ͚ 2 ͚ 3 Use Euler’s result to find the sum of the series. n෇1 n n෇1 n ϩ 1 4 ϱ 3 ϱ 1 ϱ ϱ ͩ ͪ ͚ 1 3n Ϫ 4 (a) ͚ (b) 4 17. 18. n෇1 n k෇5 ͑k Ϫ 2͒ ͚ 2 ͚ 2 n෇1 n ϩ 4 n෇3 n Ϫ 2n ͸ϱ 4 36. (a) Find the partial sum s10 of the series n෇1 1͞n . Estimate the ϱ ln n ϱ 1 19. ͚ 20. ͚ error in using s10 as an approximation to the sum of the 3 2 ϩ ϩ n෇1 n n෇1 n 6n 13 series. (b) Use 3 with n ෇ 10 to give an improved estimate of the ϱ 1 ϱ 1 21. 22. sum. ͚ ͚ 2 n෇2 n ln n n෇2 n͑ln n͒ (c) Compare your estimate in part (b) with the exact value given in Exercise 35. ϱ e1͞n ϱ n2 23. 24. (d) Find a value of ns so that n is within 0.00001 of the sum. ͚ 2 ͚ n n෇1 n n෇3 e 37. (a) Use the sum of the first 10 terms to estimate the sum of the ϱ ϱ ͸ϱ 2 1 n series n෇1 1͞n . How good is this estimate? 25. ͚ 26. ͚ 2 3 4 (b) Improve this estimate using 3 with n ෇ 10 . n෇1 n ϩ n n෇1 n ϩ 1 (c) Compare your estimate in part (b) with the exact value given in Exercise 34. 27–28 Explain why the Integral Test can’t be used to determine (d) Find a value of n that will ensure that the error in the whether the series is convergent. approximation s Ϸ sn is less than 0.001 . ϱ ␲ ϱ 2 cos n cos n ͸ϱ 5 27. ͚ 28. ͚ 38. Find the sum of the series n෇1 1͞n correct to three decimal s ϩ 2 n෇1 n n෇1 1 n places.

ϱ Ϫ6 39. Estimate ͸ ෇ ͑2n ϩ 1͒ correct to ﬁve decimal places. 29–32 Find the values of p for which the series is convergent. n 1

ϱ ϱ ϱ 2 1 1 40. ͸ ෇ ͓͞ ͑ ͒ ͔ 29. 30. How many terms of the series n 2 1 n ln n would you ͚ p ͚ p n෇2 n͑ln n͒ n෇3 n ln n ͓ln͑ln n͔͒ need to add to ﬁnd its sum to within 0.01 ?

ϱ ϱ 41. 2 p ln n Show that if we want to approximate the sum of the series 31. ͚ n͑1 ϩ n ͒ 32. ͚ ϱ Ϫ1.001 p ͸ ෇ n෇1 n෇1 n n 1 n so that the error is less than 5 in the ninth decimal place, then we need to add more than 1011,301 terms!

33. ␨ ͸ϱ 2 2 The Riemann zeta-function is deﬁned by CAS 42. (a) Show that the series n෇1 ͑ln n͒ ͞n is convergent. ϱ 1 (b) Find an upper bound for the error in the approximation ␨͑ ͒ ෇ Ϸ x ͚ x s sn. ෇ n n 1 (c) What is the smallest value of n such that this upper bound and is used in number theory to study the distribution of prime is less than 0.05 ?

numbers. What is the domain of ␨ ? (d) Find sn for this value of n .

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722 CHAPTER 11 INFINITE SEQUENCES AND SERIES

43. (a) Use 4 to show that if sn is the nth partial sum of the har- (b) Interpret monic series, then Ϫ ෇ ͓ ͑ ϩ ͒ Ϫ ͔ Ϫ 1 ഛ ϩ tn tnϩ1 ln n 1 ln n sn 1 ln n n ϩ 1 (b) The harmonic series diverges, but very slowly. Use part (a) as a difference of areas to show that t Ϫ t ϩ Ͼ 0. There- to show that the sum of the ﬁrst million terms is less than n n 1 fore ͕t ͖ is a decreasing sequence. 15 and the sum of the ﬁrst billion terms is less than 22. n (c) Use the Monotonic Sequence Theorem to show that ͕tn ͖ is 44. Use the following steps to show that the sequence convergent. ͸ϱ ln n 1 1 1 45. Find all positive values of b for which the series n෇1 b tn ෇ 1 ϩ ϩ ϩиииϩ Ϫ ln n 2 3 n converges. has a limit. (The value of the limit is denoted by ␥ and is called 46. Find all values of c for which the following series converges. Euler’s constant.) ͑ ͒ ෇ ͞ ϱ (a) Draw a picture like Figure 6 with f x 1 x and interpret c Ϫ 1 Ͼ ͚ͩ ͪ tn as an area [or use 5 ] to show that tn 0 for all n . n෇1 n n ϩ 1

11.4 The Comparison Tests

In the comparison tests the idea is to compare a given series with a series that is known to be convergent or divergent. For instance, the series

ϱ 1 1 ͚ n n෇1 2 ϩ 1

͸ϱ ͞ n ෇ 1 ෇ 1 reminds us of the series n෇1 1 2 , which is a geometric series with a 2 and r 2 and is therefore convergent. Because the series 1 is so similar to a convergent series, we have the feeling that it too must be convergent. Indeed, it is. The inequality

1 1 Ͻ 2n ϩ 1 2n

shows that our given series 1 has smaller terms than those of the geometric series and therefore all its partial sums are also smaller than 1 (the sum of the geometric series). This means that its partial sums form a bounded increasing sequence, which is convergent. It also follows that the sum of the series is less than the sum of the geometric series:

ϱ 1 Ͻ ͚ n 1 n෇1 2 ϩ 1

Similar reasoning can be used to prove the following test, which applies only to series whose terms are positive. The ﬁrst part says that if we have a series whose terms are smaller than those of a known convergent series, then our series is also convergent. The second part says that if we start with a series whose terms are larger than those of a known divergent series, then it too is divergent.

͸ ͸ The Comparison Test Suppose that an and bn are series with positive terms. ͸ ͸ (i) If bn is convergent and an ഛ bn for all n , then an is also convergent. ͸ ͸ (ii) If bn is divergent and an ജ bn for all n , then an is also divergent.

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SECTION 11.4 THE COMPARISON TESTS 723

It is important to keep in mind the distinction PROOF between a sequence and a series. A sequence n n ϱ is a list of numbers, whereas a series is a sum. ෇ ෇ ෇ ͸ (i) Let sn ͚ ai tn ͚ bi t ͚ bn With every series an there are associated two i෇1 i෇1 n෇1

sequences: the sequence ͕an ͖ of terms and the sequence ͕s ͖ of partial sums. n Since both series have positive terms, the sequences ͕sn ͖͕ and tn ͖ are increasing

͑snϩ1 ෇ sn ϩ anϩ1 ജ sn ͒. Also tn l tt , so n ഛ tna for all . Since i ഛ bi, we have sn ഛ tn. Thus s ഛ tn for all . This means that ͕s ͖ is increasing and bounded above and therefore n n ͸ converges by the Monotonic Sequence Theorem. Thus an converges. (ii) If ͸ b is divergent, then t l ϱ (since ͕t ͖ is increasing). But a ജ b so s ജ t . n ͸ n n i i n n Thus sn l ϱ . Therefore an diverges.

͸ Standard Series for Use In using the Comparison Test we must, of course, have some known series bn for the with the Comparison Test purpose of comparison. Most of the time we use one of these series:

■ A p-series [͸ 1͞n p con verges if p Ͼ 1 and diverges if p ഛ 1 ; see (11.3.1)] Ϫ ■ A geometric series [͸ ar n 1 converges if Խ r Խ Ͻ 1 and diverges if Խ r Խ ജ 1 ; see (11.2.4)]

ϱ 5 v EXAMPLE 1 Determine whether the series ͚ 2 converges or diverges. n෇1 2n ϩ 4n ϩ 3 SOLUTION For large n the dominant term in the denominator is 2n2 , so we compare the given series with the series ͸ 5͑͞2n2 ͒ . Observe that 5 5 Ͻ 2n2 ϩ 4n ϩ 3 2n2

because the left side has a bigger denominator. (In the notation of the Comparison Test,

an is the left side and bn is the right side.) We know that

ϱ ϱ 5 ෇ 5 1 ͚ 2 ͚ 2 n෇1 2n 2 n෇1 n is convergent because it’s a constant times a pp -series with ෇ 2 Ͼ 1 . Therefore ϱ 5 ͚ 2 n෇1 2n ϩ 4n ϩ 3 is convergent by part (i) of the Comparison Test.

NOTE 1 Although the condition an ഛ bn or an ജ bn in the Comparison Test is given for all nn , we need verify only that it holds for ജ NN , where is some ﬁxed integer, because the convergence of a series is not affected by a ﬁnite number of terms. This is illustrated in the next example.

ϱ ln k v EXAMPLE 2 Test the series ͚ for convergence or divergence. k෇1 k SOLUTION We used the Integral Test to test this series in Example 4 of Section 11.3, but we can also test it by comparing it with the harmonic series. Observe that ln k Ͼ 1 for k ജ 3 and so ln k 1 Ͼ k ജ 3 k k

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724 CHAPTER 11 INFINITE SEQUENCES AND SERIES

We know that ͸ 1͞k is divergent (p -series with p ෇ 1 ). Thus the given series is divergent by the Comparison Test.

NOTE 2 The terms of the series being tested must be smaller than those of a convergent series or larger than those of a divergent series. If the terms are larger than the terms of a convergent series or smaller than those of a divergent series, then the Comparison Test doesn’t apply. Consider, for instance, the series

ϱ 1 ͚ n n෇1 2 Ϫ 1 The inequality 1 Ͼ 1 2n Ϫ 1 2n

͸ ෇ ͸ 1 n is useless as far as the Comparison Test is concerned because bn ( 2 ) is convergent ͸ n and an Ͼ bn . Nonetheless, we have the feeling that 1͑͞2 Ϫ 1͒ ought to be convergent ͸ 1 n because it is very similar to the convergent geometric series ( 2 ) . In such cases the following test can be used.

͸ ͸ The Limit Comparison Test Suppose that an and bn are series with positive terms. If Exercises 40 and 41 deal with the an lim ෇ c cases ෇ and ෇ ϱ. l ϱ c 0 c n bn

where c is a ﬁnite number and c Ͼ 0 , then either both series converge or both diverge.

PROOF Let m and M be positive numbers such that m Ͻ c Ͻ Ma . Because n͞bn is close to c for large n, there is an integer N such that

a m Ͻ n Ͻ M when n Ͼ N bn

and so mbn Ͻ an Ͻ Mbn when n Ͼ N

If ͸ b converges, so does ͸ Mb . Thus ͸ a converges by part (i) of the Comparison ͸n ͸ n n ͸ Test. If bn diverges, so does mbn and part (ii) of the Comparison Test shows that an diverges.

ϱ 1 EXAMPLE 3 Test the series ͚ n for convergence or divergence. n෇1 2 Ϫ 1 SOLUTION We use the Limit Comparison Test with

1 1 a ෇ b ෇ n 2n Ϫ 1 n 2n and obtain

a 1͑͞2 n Ϫ 1͒ 2n 1 lim n ෇ lim ෇ lim ෇ lim ෇ 1 Ͼ 0 l ϱ l ϱ n l ϱ n l ϱ n n bn n 1͞2 n 2 Ϫ 1 n 1 Ϫ 1͞2

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SECTION 11.4 THE COMPARISON TESTS 725

Since this limit exists and ͸ 1͞2n is a convergent geometric series, the given series converges by the Limit Comparison Test.

ϱ 2n2 ϩ 3n EXAMPLE 4 Determine whether the series ͚ converges or diverges. 5 n෇1 s5 ϩ n SOLUTION The dominant part of the numerator is 2n2 and the dominant part of the denominator is sn5 ෇ n 5͞2 . This suggests taking

2n2 ϩ 3n 2n2 2 an ෇ bn ෇ ෇ s5 ϩ n 5 n 5͞2 n 1͞2

a 2n2 ϩ 3n n 1͞2 2n5͞2 ϩ 3n3͞2 lim n ෇ lim ؒ ෇ lim l ϱ l ϱ 5 l ϱ 5 n bn n s5 ϩ n 2 n 2s5 ϩ n 3 2 ϩ n 2 ϩ 0 ෇ lim ෇ ෇ 1 n l ϱ 5 2s0 ϩ 1 2ͱ ϩ 1 n5 ͸ ෇ ͸ ͞ 1͞2 ෇ 1 Ͻ Since bn 2 1 n is divergent (pp-series with 2 1), the given series diverges by the Limit Comparison Test. ͸ Notice that in testing many series we ﬁnd a suitable comparison series bn by keeping only the highest powers in the numerator and denominator. Estimating Sums If we have used the Comparison Test to show that a series ͸ a converges by comparison ͸ ͸ n with a series bn , then we may be able to estimate the sum an by comparing remainders. As in Section 11.3, we consider the remainder

Rn ෇ s Ϫ sn ෇ anϩ1 ϩ anϩ2 ϩиии ͸ For the comparison series bn we consider the corresponding remainder

Tn ෇ t Ϫ tn ෇ bnϩ1 ϩ bnϩ2 ϩиии

Since a ഛ b for all nR , we have ഛ T . If ͸ b is a p -series, we can estimate its remain- n n ͸ n n n der Tn as in Section 11.3. If bn is a geometric series, then Tn is the sum of a geometric series and we can sum it exactly (see Exercises 35 and 36). In either case we know that Rn is smaller than Tn .

v EXAMPLE 5 Use the sum of the ﬁrst 100 terms to approximate the sum of the series ͸ 1͑͞n3 ϩ 1͒. Estimate the error involved in this approximation. SOLUTION Since 1 1 Ͻ n3 ϩ 1 n3

the given series is convergent by the Comparison Test. The remainder Tn for the comparison series ͸ 1͞n3 was estimated in Example 5 in Section 11.3 using the Remainder Esti- mate for the Integral Test. There we found that

ϱ 1 1 Tn ഛ y dx ෇ n x 3 2n2

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726 CHAPTER 11 INFINITE SEQUENCES AND SERIES

Therefore the remainder Rn for the given series satisﬁes 1 R ഛ T ഛ n n 2n2 With n ෇ 100 we have 1 R ഛ ෇ 0.00005 100 2͑100͒2 Using a programmable calculator or a computer, we ﬁnd that

ϱ 100 1 Ϸ 1 Ϸ ͚ 3 ͚ 3 0.6864538 n෇1 n ϩ 1 n෇1 n ϩ 1 with error less than 0.00005 .

11.4 Exercises

1. ͸ ͸ ͸ ϱ ϱ Suppose an and bn are series with positive terms and bn sn ϩ 2 n ϩ 2 21. ͚ 22. ͚ is known to be convergent. 2 ϩ ϩ ͑ ϩ ͒ 3 ͸ n෇1 2n n 1 n෇3 n 1 (a) If an Ͼ bn for all n , what can you say about an ? Why? ͸ ϱ ϱ 2 (b) If an Ͻ bn for all n , what can you say about an ? Why? 5 ϩ 2n n Ϫ 5n 23. ͚ 24. ͚ ͑ ϩ 2͒2 3 ϩ ϩ ͸ ͸ ͸ n෇1 1 n n෇1 n n 1 2. Suppose an and bn are series with positive terms and bn is known to be divergent. ϱ sn4 ϩ 1 ϱ 1 (a) If a Ͼ b for all n, what can you say about ͸ a ? Why? 25. ͚ 26. ͚ n n n 3 ϩ 2 s 2 Ϫ ͸ n෇1 n n n෇2 n n 1 (b) If an Ͻ bn for all n, what can you say about an ? Why? ϱ 2 ϱ 1͞n 3–32 Determine whether the series converges or diverges. 1 Ϫ e 27. ͚ ͩ1 ϩ ͪ e n 28. ͚ ϱ ϱ ෇ n ෇ n n n3 n 1 n 1 3. ͚ 4. ͚ 3 ϩ 4 Ϫ ϱ ϱ n෇1 2n 1 n෇2 n 1 1 n! 29. 30. ͚ ͚ n ϱ ϱ ෇ n! ෇ n n ϩ 1 n Ϫ 1 n 1 n 1 5. ͚ 6. ͚ s 2s ϱ ϱ n෇1 n n n෇1 n n 1 1 31. ͩ ͪ 32. ͚ ͚ sin 1ϩ1͞n ϱ 9n ϱ 6n n෇1 n n෇1 n 7. 8. ͚ n ͚ n n෇1 3 ϩ 10 n෇1 5 Ϫ 1

ϱ ln k ϱ k sin2k 33–36 9. 10. Use the sum of the ﬁrst 10 terms to approximate the sum of ͚ ͚ 3 k෇1 k k෇1 1 ϩ k the series. Estimate the error. ϱ ϱ 2 ϱ ϱ 1 sin n s3 k ͑2k Ϫ 1͒͑k 2 Ϫ 1͒ 33. 34. ͚ ͚ 3 11. ͚ 12. ͚ ෇ 4 ϩ ෇ 3 2 2 n 1 sn 1 n 1 n k෇1 sk ϩ 4k ϩ 3 k෇1 ͑k ϩ 1͒͑k ϩ 4͒ ϱ ϱ ϱ ϱ 1 arctan n sn 35. Ϫn 2 36. 13. 14. ͚ 5 cos n ͚ n n ͚ 1.2 ͚ n෇1 n෇1 3 ϩ 4 n෇1 n n෇2 n Ϫ 1

ϱ 4nϩ1 ϱ 1 15. ͚ 16. ͚ 37. n 3 4 The meaning of the decimal representation of a number n෇1 3 Ϫ 2 n෇1 s3n ϩ 1 0.d1d2d3 ...(where the digit di is one of the numbers 0, 1, ϱ 1 ϱ 1 2, . . . , 9) is that 17. ͚ 18. ͚ 2 n෇1 sn ϩ 1 n෇1 2n ϩ 3 ෇ d1 ϩ d2 ϩ d3 ϩ d4 ϩиии 0.d1d2d3d4 ... 2 3 4 ϱ 1 ϩ 4n ϱ n ϩ 4n 10 10 10 10 19. ͚ 20. ͚ ϩ n ϩ n n෇1 1 3 n෇1 n 6 Show that this series always converges.

1. Homework Hints available at stewartcalculus.com

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SECTION 11.5 ALTERNATING SERIES 727

͸ϱ p 38. For what values of p does the series n෇2 1͑͞n ln n͒ converge? (b) Use part (a) to show that the series diverges. ϱ ϱ 39. ജ ͸ ͸ 2 1 ln n Prove that if an 0 and an converges, then an also (i) ͚ (ii) ͚ converges. n෇2 ln n n෇1 n 40. ͸ ͸ (a) Suppose that an and bn are series with positive terms 42. ͸ ͸ ͸ Give an example of a pair of series an and bn with positive and bn is convergent. Prove that if ͸ ͸ terms where lim n l ϱ ͑an͞bn͒ ෇ 0 and bn diverges, but an a lim n ෇ 0 converges. (Compare with Exercise 40.) l ϱ n bn 43. Ͼ ͸ ͸ Show that if an 0 and lim n l ϱ nan 0, then an is then an is also convergent. (b) Use part (a) to show that the series converges. divergent. ϱ ϱ ln n ln n 44. Ͼ ͸ ͸ ͑ ϩ ͒ (i) ͚ (ii) ͚ Show that if an 0 and an is convergent, then ln 1 an 3 s n n෇1 n n෇1 n e is convergent. ͸ ͸ 41. (a) Suppose that an and bn are series with positive terms ͸ 45. ͸ and bn is divergent. Prove that if If an is a convergent series with positive terms, is it true that ͸ sin͑a ͒ is also convergent? a n lim n ෇ ϱ n l ϱ b n 46. If ͸ a and ͸ b are both convergent series with positive terms, ͸ n ͸n then an is also divergent. is it true that an bn is also convergent?

11.5 Alternating Series

The convergence tests that we have looked at so far apply only to series with positive terms. In this section and the next we learn how to deal with series whose terms are not necessarily positive. Of particular importance are alternating series, whose terms alternate in sign. An alternating series is a series whose terms are alternately positive and negative. Here are two examples: 1 1 1 1 1 ϱ 1 1 Ϫ ϩ Ϫ ϩ Ϫ ϩиии෇ ͚ ͑Ϫ1͒nϪ1 2 3 4 5 6 n෇1 n

1 2 3 4 5 6 ϱ n Ϫ ϩ Ϫ ϩ Ϫ ϩ Ϫиии෇ ͚ ͑Ϫ1͒n 2 3 4 5 6 7 n෇1 n ϩ 1 We see from these examples that the nth term of an alternating series is of the form

nϪ1 n an ෇ ͑Ϫ1͒ bn or an ෇ ͑Ϫ1͒ bn

where bn is a positive number. (In fact, bn ෇ Խ a n Խ .) The following test says that if the terms of an alternating series decrease toward 0 in absolute value, then the series converges.

Alternating Series Test If the alternating series

ϱ nϪ1 ͚ ͑Ϫ1͒ bn ෇ b1 Ϫ b2 ϩ b3 Ϫ b4 ϩ b5 Ϫ b6 ϩиии bn Ͼ 0 n෇1 satisﬁes

(i) bnϩ1 ഛ bn for all n

(ii) lim bn ෇ 0 n l ϱ then the series is convergent.

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728 CHAPTER 11 INFINITE SEQUENCES AND SERIES

Before giving the proof let’s look at Figure 1, which gives a picture of the idea behind

the proof. We first plot s1 ෇ b1 on a number line. To find s2 we subtract b2 , so s2 is to the left of s1 . Then to find s3 we add b3 , so s3 is to the right of s2 . But, since b3 Ͻ b2 , s3 is to the left of s1 . Continuing in this manner, we see that the partial sums oscillate back and forth. Since bn l 0 , the successive steps are becoming smaller and smaller. The even par-

tial sums s2 , s4 , s6 , . . . are increasing and the odd partial sums s1 , s3 , s5 , . . . are decreasing. Thus it seems plausible that both are converging to some number s , which is the sum of the series. Therefore we consider the even and odd partial sums separately in the following proof. b¡ -b™ +b£ -b¢ +b∞ -bß

FIGURE 1 0 s™ s¢ sß s s∞ s£ s¡

PROOF OF THE ALTERNATING SERIES TEST We ﬁrst consider the even partial sums:

s2 ෇ b1 Ϫ b2 ജ 0 since b2 ഛ b1

s4 ෇ s2 ϩ ͑b3 Ϫ b4 ͒ ജ s2 since b4 ഛ b3

In general s2n ෇ s2nϪ2 ϩ ͑b2nϪ1 Ϫ b2n ͒ ജ s2nϪ2 since b2n ഛ b2nϪ1

Thus 0 ഛ s2 ഛ s4 ഛ s6 ഛиииഛs2n ഛиии

But we can also write

s2n ෇ b1 Ϫ ͑b2 Ϫ b3 ͒ Ϫ ͑b4 Ϫ b5 ͒ ϪиииϪ͑b2nϪ2 Ϫ b2nϪ1͒ Ϫ b2n

Every term in brackets is positive, so s2n ഛ b1 for all n . Therefore the sequence ͕s2n ͖ of even partial sums is increasing and bounded above. It is therefore convergent by the Monotonic Sequence Theorem. Let’s call its limit s , that is,

lim s2n ෇ s n l ϱ

Now we compute the limit of the odd partial sums:

lim s2nϩ1 ෇ lim ͑s2n ϩ b2nϩ1͒ n l ϱ n l ϱ

෇ lim s2n ϩ lim b2nϩ1 n l ϱ n l ϱ

෇ s ϩ 0 [by condition (ii)] ෇ s

Since both the even and odd partial sums converge to s , we have lim n l ϱ sn ෇ s [see Exercise 92(a) in Section 11.1] and so the series is convergent.

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SECTION 11.5 ALTERNATING SERIES 729

Figure 2 illustrates Example 1 by showing the v EXAMPLE 1 The alternating harmonic series nϪ1 graphs of the terms an ෇ ͑Ϫ1͒ ͞n and the partial sums . Notice how the values of ϱ Ϫ sn sn 1 1 1 ͑Ϫ1͒n 1 zigzag across the limiting value, which appears 1 Ϫ ϩ Ϫ ϩиии෇ ͚ to be about 0.7 . In fact, it can be proved that 2 3 4 n෇1 n the exact sum of the series is ln 2 Ϸ 0.693 (see Exercise 36). satisﬁes 1 1 (i) b ϩ Ͻ b because Ͻ n 1 n n ϩ 1 n 1 1 ͕s ͖ (ii) lim bn ෇ lim ෇ 0 n n l ϱ n l ϱ n so the series is convergent by the Alternating Series Test.

ϱ ͑Ϫ ͒n ͕a ͖ 1 3n n v EXAMPLE 2 The series ͚ is alternating, but n෇1 4n Ϫ 1 0 n 3n 3 3 lim bn ෇ lim ෇ lim ෇ n l ϱ n l ϱ 4n Ϫ 1 n l ϱ 1 4 4 Ϫ n

FIGURE 2 so condition (ii) is not satisﬁed. Instead, we look at the limit of thenth term of the series:

͑Ϫ1͒n3n lim an ෇ lim n l ϱ n l ϱ 4n Ϫ 1 This limit does not exist, so the series diverges by the Test for Divergence.

ϱ n2 EXAMPLE 3 ͑Ϫ ͒nϩ1 Test the series ͚ 1 3 for convergence or divergence. n෇1 n ϩ 1 SOLUTION The given series is alternating so we try to verify conditions (i) and (ii) of the Alternating Series Test. Unlike the situation in Example 1, it is not obvious that the sequence given by 2 3 bn ෇ n ͑͞n ϩ 1͒ is decreasing. However, if we consider the related function f ͑x͒ ෇ x 2͑͞x 3 ϩ 1͒, we ﬁnd that x͑2 Ϫ x 3 ͒ f Ј͑x͒ ෇ ͑x 3 ϩ 1͒2

Since we are considering only positive xf , we see that Ј͑x͒ Ͻ 02 if Ϫ x 3 Ͻ 0 , that is, x Ͼ s3 2 . Thus f is decreasing on the interval (s3 2 , ϱ). This means that f ͑n ϩ 1͒ Ͻ f ͑n͒ Instead of verifying condition (i) of the Alter- and therefore bnϩ1 Ͻ bn when n ജ 2 . (The inequality b2 Ͻ b1 can be veriﬁed directly but nating Series Test by computing a derivative, all that really matters is that the sequence ͕bn ͖ is eventually decreasing.) we could verify that bnϩ1 Ͻ bn directly by using the techni que of Solution1 of Condition (ii) is readily veriﬁed: Example 13 in Section 11.1. 1 n2 n lim bn ෇ lim ෇ lim ෇ 0 n l ϱ n l ϱ n3 ϩ 1 n l ϱ 1 1 ϩ n3

Thus the given series is convergent by the Alternating Series Test.

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730 CHAPTER 11 INFINITE SEQUENCES AND SERIES

Estimating Sums

A partial sum sn of any convergent series can be used as an approximation to the total sum s, but this is not of much use unless we can estimate the accuracy of the approximation. The

error involved in using s Ϸ sn is the remainder Rn ෇ s Ϫ sn . The next theorem says that for series that satisfy the conditions of the Alternating Series Test, the size of the error is

smaller than bnϩ1 , which is the absolute value of the ﬁrst neglected term.

͸ nϪ1 Alternating Series Estimation Theorem If s ෇ ͑Ϫ1͒ bn is the sum of an alter nating You can see geometrically why the Alternating series that satisﬁes Series Estimation Theorem is true by looking at Figure 1 (on page 728). Notice that Ϫ s s4 Ͻ b5, (i) bnϩ1 ഛ bn and (ii) lim bn ෇ 0 n l ϱ Խ s Ϫ s5 Խ Ͻ b6, and so on. Notice also that s lies between any two consecutive partial sums. then Խ Rn Խ ෇ Խ s Ϫ sn Խ ഛ bnϩ1

PROOF We know from the proof of the Alternating Series Test that s lies between any two

consecutive partial sums sn and snϩ1 . (There we showed that s is larger than all even partial sums. A similar argument shows that s is smaller than all the odd sums.) It follows that

Խ s Ϫ sn Խ ഛ Խ snϩ1 Ϫ sn Խ ෇ bnϩ1

ϱ ͑Ϫ1͒n By definition, 0! ෇ 1 . v EXAMPLE 4 Find the sum of the series ͚ correct to three decimal places. n෇0 n! SOLUTION We first observe that the series is convergent by the Alternating Series Test because 1 1 1 (i) ෇ Ͻ ͑n ϩ 1͒! n! ͑n ϩ 1͒ n! 1 1 1 (ii) 0 Ͻ Ͻ l 0 so l 0 as n l ϱ n! n n! To get a feel for how many terms we need to use in our approximation, let’s write out the first few terms of the series: 1 1 1 1 1 1 1 1 s ෇ Ϫ ϩ Ϫ ϩ Ϫ ϩ Ϫ ϩиии 0! 1! 2! 3! 4! 5! 6! 7! ෇ Ϫ ϩ 1 Ϫ 1 ϩ 1 Ϫ 1 ϩ 1 Ϫ 1 ϩиии 1 1 2 6 24 120 720 5040

෇ 1 Ͻ 1 ෇ Notice that b7 5040 5000 0.0002

෇ Ϫ ϩ 1 Ϫ 1 ϩ 1 Ϫ 1 ϩ 1 Ϸ and s6 1 1 2 6 24 120 720 0.368056

By the Alternating Series Estimation Theorem we know that

Ϫ ഛ Ͻ In Section 11.10 we will prove that Խ s s6 Խ b7 0.0002 x ͸ϱ n e ෇ n෇0 x ͞n! for all x , so what we have obtained in Example 4 is actually an This error of less than 0.0002 does not affect the third decimal place, so we have approximation to the number e Ϫ1 . s Ϸ 0.368 correct to three decimal places.

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SECTION 11.5 ALTERNATING SERIES 731

| NOTE The rule that the error (in using sn to approximate s ) is smaller than the ﬁrst neglected term is, in general, valid only for alternating series that satisfy the conditions of the Alternating Series Estimation Theorem. The rule does not apply to other types of series.

11.5 Exercises

1. (a) What is an alternating series? ϱ n 22. ͑Ϫ ͒nϪ1 ͚ 1 n (b) Under what conditions does an alternating series n෇1 8 converge? (c) If these conditions are satisﬁed, what can you say about the remainder after n terms? 23–26 Show that the series is convergent. How many terms of the series do we need to add in order to ﬁnd the sum to the indi- 2–20 Test the series for convergence or divergence. cated accuracy? 2. 2 Ϫ 2 ϩ 2 Ϫ 2 ϩ 2 Ϫиии ϱ ͑Ϫ1͒nϩ1 3 5 7 9 11 23. Ͻ ͚ 6 (Խ error Խ 0.00005) ෇ n 3. Ϫ2 ϩ 4 Ϫ 6 ϩ 8 Ϫ 10 ϩиии n 1 5 6 7 8 9 ϱ ͑Ϫ1͒n 1 1 1 1 1 24. Ͻ 4. Ϫ ϩ Ϫ ϩ Ϫиии ͚ n (Խ error Խ 0.0001) ෇ n s2 s3 s4 s5 s6 n 1 5

ϱ n ϱ nϪ1 ϱ nϪ1 ͑Ϫ1͒ ͑Ϫ1͒ ͑Ϫ1͒ 25. Խ Խ Ͻ 5. ͚ 6. ͚ ͚ n ( error 0.000005) n෇0 10 n! n෇1 2n ϩ 1 n෇1 ln͑n ϩ 4͒

ϱ ϱ ϱ 3n Ϫ 1 n nϪ1 Ϫn 7. ͚ ͑Ϫ1͒n 8. ͚ ͑Ϫ1͒n 26. ͚ ͑Ϫ1͒ ne (Խ error Խ Ͻ 0.01) ෇ n෇1 2n ϩ 1 n෇1 sn3 ϩ 2 n 1

ϱ ϱ sn 9. ͑Ϫ ͒n Ϫn 10. ͑Ϫ ͒n ͚ 1 e ͚ 1 27–30 n෇1 n෇1 2n ϩ 3 Approximate the sum of the series correct to four decimal places. ϱ n2 ϱ 11. ͚ ͑Ϫ1͒nϩ1 12. ͚ ͑Ϫ1͒nϩ1neϪn ϱ ͑Ϫ1͒n ϱ ͑Ϫ1͒nϩ1 3 ϩ 27. 28. n෇1 n 4 n෇1 ͚ ͚ 6 n෇1 ͑2n͒! n෇1 n ϱ ϱ 13. ͚ ͑Ϫ1͒nϪ1e 2͞n 14. ͚ ͑Ϫ1͒nϪ1 arctan n ϱ ͑Ϫ1͒nϪ1 n2 ϱ ͑Ϫ1͒n ෇ ෇ 29. 30. n 1 n 1 ͚ n ͚ n n෇1 10 n෇1 3 n! ϱ sin n ϩ 1 ␲ ϱ n cos n␲ 15. ( 2 ) 16. ͚ ͚ n n෇0 1 ϩ sn n෇1 2 31. Is the 50th partial sum s50 of the alternating series ͸ϱ nϪ1 ϱ ␲ ϱ ␲ n෇1 ͑Ϫ1͒ ͞n an overestimate or an underestimate of the 17. ͚ ͑Ϫ1͒n sinͩ ͪ 18. ͚ ͑Ϫ1͒n cosͩ ͪ total sum? Explain. n෇1 n n෇1 n 32–34 ϱ n n ϱ For what values of p is each series convergent? 19. ͑Ϫ ͒n 20. ͑Ϫ ͒n s ϩ Ϫ s ͚ 1 ͚ 1 ( n 1 n ) ϱ nϪ1 ෇ ෇ ͑Ϫ1͒ n 1 n! n 1 32. ͚ p n෇1 n

; 21–22 Graph both the sequence of terms and the sequence of ϱ ͑Ϫ ͒n ϱ ͑ ͒ p 1 nϪ1 ln n partial sums on the same screen. Use the graph to make a rough 33. ͚ 34. ͚ ͑Ϫ1͒ n෇1 n ϩ p n෇2 n estimate of the sum of the series. Then use the Alternating Series Estimation Theorem to estimate the sum correct to four decimal places. ͸ nϪ1 35. Show that the series ͑Ϫ1͒ bn , where bn ෇ 1͞nn if is odd ϱ ͑Ϫ0.8͒n ෇ ͞ 2 21. ͚ and bn 1 n if n is even, is divergent. Why does the Alter- n෇1 n! nating Series Test not apply?

; Graphing calculator or computer required 1. Homework Hints available at stewartcalculus.com

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732 CHAPTER 11 INFINITE SEQUENCES AND SERIES

36. Use the following steps to show that (b) From Exercise 44 in Section 11.3 we have h Ϫ ln n l ␥ as n l ϱ ϱ ͑Ϫ1͒nϪ1 n ͚ ෇ ln 2 n෇1 n and therefore

h2n Ϫ ln͑2n͒ l ␥ as n l ϱ Let hn and sn be the partial sums of the harmonic and alternating harmonic series. Use these facts together with part (a) to show that

(a) Show that s2n ෇ h2n Ϫ hn. s2n l ln 2 as n l ϱ.

11.6 Absolute Convergence and the Ratio and Root Tests ͸ Given any series an , we can consider the corresponding series

͚ Խ an Խ ෇ Խ a1 Խ ϩ Խ a2 Խ ϩ Խ a3 Խ ϩиии n෇1

whose terms are the absolute values of the terms of the original series.

͸ We have convergence tests for series with 1 Deﬁnition A series an is called absolutely convergent if the series of positive terms and for alternating series. But ͸ absolute values Խ an Խ is convergent. what if the signs of the terms switch back and forth irregularly? We will see in Example 3 that the idea of absolute convergence sometimes ͸ helps in such cases. Notice that if an is a series with positive terms, then Խ an Խ ෇ an and so absolute convergence is the same as convergence in this case.

EXAMPLE 1 The series

ϱ ͑Ϫ ͒nϪ1 1 ෇ Ϫ 1 ϩ 1 Ϫ 1 ϩиии ͚ 2 1 2 2 2 n෇1 n 2 3 4

is absolutely convergent because

ϱ ͑Ϫ ͒nϪ1 ϱ ͚ Ϳ 1 Ϳ ෇ ͚ 1 ෇ ϩ 1 ϩ 1 ϩ 1 ϩиии 2 2 1 2 2 2 n෇1 n n෇1 n 2 3 4

is a convergent p -series (p ෇ 2 ).

EXAMPLE 2 We know that the alternating harmonic series

ϱ ͑Ϫ1͒nϪ1 1 1 1 ͚ ෇ 1 Ϫ ϩ Ϫ ϩиии n෇1 n 2 3 4

is convergent (see Example 1 in Section 11.5), but it is not absolutely convergent because the corresponding series of absolute values is

ϱ ͑Ϫ1͒nϪ1 ϱ 1 1 1 1 ͚ Ϳ Ϳ ෇ ͚ ෇ 1 ϩ ϩ ϩ ϩиии n෇1 n n෇1 n 2 3 4

which is the harmonic series (p -series with p ෇ 1 ) and is therefore divergent.

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SECTION 11.6 ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS 733

͸ 2 Deﬁnition A series an is called conditionally convergent if it is convergent but not absolutely convergent.

Example 2 shows that the alternating harmonic series is conditionally convergent. Thus it is possible for a series to be convergent but not absolutely convergent. However, the next theorem shows that absolute convergence implies convergence.

͸ 3 Theorem If a series an is absolutely convergent, then it is convergent.

PROOF Observe that the inequality

0 ഛ an ϩ Խ an Խ ഛ 2Խ an Խ ͸ ͸ is true because Խ an Խ is either an or Ϫan . If an is absolutely convergent, then Խ a n Խ is ͸ ͸ convergent, so 2Խ an Խ is convergent. Therefore, by the Comparison Test, (an ϩ Խ an Խ) is convergent. Then

͚ an ෇ ͚ (an ϩ Խ an Խ) Ϫ ͚ Խ an Խ

is the difference of two convergent series and is therefore convergent.

v EXAMPLE 3 Determine whether the series

ϱ cos n ෇ cos 1 ϩ cos 2 ϩ cos 3 ϩиии ͚ 2 2 2 2 n෇1 n 1 2 3

is convergent or divergent.

Figure 1 shows the graphs of the terms an and SOLUTION This series has both positive and negative terms, but it is not alternating. partial sums of the series in Example 3. sn (The ﬁrst term is positive, the next three are negative, and the following three are posi- Notice that the series is not alternating but has positive and negative terms. tive: The signs change irregularly.) We can apply the Comparison Test to the series of absolute values ϱ cos n ϱ Խ cos n Խ 0.5 ͚ Ϳ Ϳ ෇ ͚ 2 2 ͕ ͖ n෇1 n n෇1 n sn Since Խ cos n Խ ഛ 1 for all n , we have

͕a ͖ Խ cos n Խ 1 n ഛ 2 2 0 n n n

We know that ͸ 1͞n2 is convergent (pp -series with ෇ 2 ) and therefore ͸ Խ cos n Խ͞n2 is FIGURE 1 convergent by the Comparison Test. Thus the given series ͸ ͑cos n͒͞n2 is absolutely convergent and therefore convergent by Theorem 3.

The following test is very useful in determining whether a given series is absolutely convergent.

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734 CHAPTER 11 INFINITE SEQUENCES AND SERIES

The Ratio Test

ϱ a ϩ Ϳ n 1 Ϳ ෇ Ͻ ͚ (i) If lim L 1, then the series an is absolutely convergent l ϱ n an n෇1 (and therefore convergent).

ϱ Ϳ anϩ1 Ϳ ෇ Ͼ Ϳ anϩ1 Ϳ ෇ ϱ ͚ (ii) If lim L 1 or lim , then the series an l ϱ l ϱ n an n an n෇1 is divergent.

a ϩ (iii) If lim Ϳ n 1 Ϳ ෇ 1, the Ratio Test is inconclusive; that is, no conclusion can n l ϱ a n ͸ be drawn about the convergence or divergence of an .

PROOF (i) The idea is to compare the given series with a convergent geometric series. Since L Ͻ 1, we can choose a number r such that L Ͻ r Ͻ 1 . Since

a ϩ lim Ϳ n 1 Ϳ ෇ L and L Ͻ r l ϱ n an

the ratio Խ anϩ1͞an Խ will eventually be less than r ; that is, there exists an integer N such that a ϩ Ϳ n 1 Ϳ Ͻ r whenever n ജ N an or, equivalently,

4 Խ anϩ1 Խ Ͻ Խ an Խ r whenever n ജ N

Putting n successively equal to N , N ϩ 1 , N ϩ 2 , . . . in 4 , we obtain

Խ aNϩ1 Խ Ͻ Խ aN Խ r

2 Խ aNϩ2 Խ Ͻ Խ aNϩ1 Խ r Ͻ Խ aN Խ r

3 Խ aNϩ3 Խ Ͻ Խ aNϩ2 Խ r Ͻ Խ aN Խ r and, in general,

k 5 Խ aNϩk Խ Ͻ Խ aN Խ r for all k ജ 1

Now the series ϱ k 2 3 ͚ Խ aN Խ r ෇ Խ aN Խ r ϩ Խ aN Խ r ϩ Խ aN Խ r ϩиии k෇1 is convergent because it is a geometric series with 0 Ͻ r Ͻ 1 . So the inequality 5 together with the Comparison Test, shows that the series ϱ ϱ

͚ Խ an Խ ෇ ͚ Խ aNϩk Խ ෇ Խ aNϩ1 Խ ϩ Խ aNϩ2 Խ ϩ Խ aNϩ3 Խ ϩиии n෇Nϩ1 k෇1

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SECTION 11.6 ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS 735

͸ϱ is also convergent. It follows that the series n෇1 Խ an Խ is convergent. (Recall that a ﬁnite ͸ number of terms doesn’t affect convergence.) Therefore an is absolutely convergent. (ii) If Խ anϩ1͞an Խl L Ͼ 1 or Խ anϩ1͞an Խl ϱ, then the ratio Խ anϩ1͞an Խ will eventually be greater than 1; that is, there exists an integer N such that

a ϩ Ϳ n 1 Ϳ Ͼ 1 whenever n ജ N an

This means that Խ anϩ1 Խ Ͼ Խ an Խ whenever n ജ N and so

lim an 0 n l ϱ ͸ Therefore an diverges by the Test for Divergence.

NOTE Part (iii) of the Ratio Test says that if limn l ϱ Խ anϩ1͞an Խ ෇ 1, the test gives no information. For instance, for the convergent series ͸ 1͞n 2 we have

1 2 2 a ϩ ͑n ϩ 1͒ n 1 Ϳ n 1 Ϳ ෇ ෇ ෇ l 1 as n l ϱ a 1 ͑n ϩ 1͒2 1 2 n ͩ1 ϩ ͪ n2 n

whereas for the divergent series ͸ 1͞n we have

1 a ϩ n ϩ 1 n 1 Ϳ n 1 Ϳ ෇ ෇ ෇ l 1 as n l ϱ a 1 n ϩ 1 1 n 1 ϩ n n The Ratio Test is usually conclusive if the n th ͸ term of the series contains an exponential or a Therefore, if limn l ϱ Խ anϩ1͞an Խ ෇ 1, the series an might converge or it might diverge. In factorial, as we will see in Examples 4 and 5. this case the Ratio Test fails and we must use some other test.

ϱ n3 EXAMPLE 4 ͑Ϫ ͒n Test the series ͚ 1 n for absolute convergence. n෇1 3

n 3 n SOLUTION We use the Ratio Test with an ෇ ͑Ϫ1͒ n ͞3 : Estimating Sums In the last three sections we used various meth- ͑Ϫ1͒nϩ1͑n ϩ 1͒3 ods for estimating the sum of a series—the nϩ1 ͑ ϩ ͒3 n method depended on which test was used to Ϳ anϩ1 Ϳ ෇ 3 ෇ n 1 ؒ 3 n 3 nϩ1 3 prove convergence. What about series for an ͑Ϫ1͒ n 3 n which the Ratio Test works? There are two 3n possibilities: If the series happens to be an alter- | | nating series, as in Example 4, then it is best to 1 n ϩ 1 3 1 1 3 1 use the methods of Section 11.5. If the terms are ෇ ͩ ͪ ෇ ͩ1 ϩ ͪ l Ͻ 1 all positive, then use the special methods 3 n 3 n 3 explained in Exercise 38. Thus, by the Ratio Test, the given series is absolutely convergent and therefore convergent.

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736 CHAPTER 11 INFINITE SEQUENCES AND SERIES

ϱ n n v EXAMPLE 5 Test the convergence of the series ͚ . n෇1 n!

n SOLUTION Since the terms an ෇ n ͞n! are positive, we don’t need the absolute value signs. ͑ ϩ ͒nϩ1 !anϩ1 ෇ n 1 ؒ n n an ͑n ϩ 1͒! n ͑ ϩ ͒͑ ϩ ͒n !෇ n 1 n 1 ؒ n ͑n ϩ 1͒n! n n n ϩ 1 n 1 n ෇ ͩ ͪ ෇ ͩ1 ϩ ͪ l e as n l ϱ n n

(see Equation 3.6.6). Since e Ͼ 1 , the given series is divergent by the Ratio Test.

NOTE Although the Ratio Test works in Example 5, an easier method is to use the Test for Divergence. Since nn n ؒ n ؒ n ؒ иии ؒ n a ෇ ෇ ജ n n n! 1 ؒ 2 ؒ 3 ؒ иии ؒ n

it follows that an does not approach 0 as n l ϱ . Therefore the given series is divergent by the Test for Divergence. The following test is convenient to apply when n th powers occur. Its proof is similar to the proof of the Ratio Test and is left as Exercise 41.

The Root Test ϱ n (i) If lim sԽ an Խ ෇ L Ͻ 1, then the series ͚ an is absolutely convergent l ϱ n n෇1 (and therefore convergent). ϱ n n (ii) If lim sԽ an Խ ෇ L Ͼ 1 or lim sԽ an Խ ෇ ϱ, then the series ͚ an is divergent. l ϱ l ϱ n n n෇1

n (iii) If lim sԽ an Խ ෇ 1, the Root Test is inconclusive. n l ϱ

n If lim n l ϱ sԽ an Խ ෇ 1, then part (iii) of the Root Test says that the test gives no infor- ͸ mation. The series an could converge or diverge. (If L ෇ 1 in the Ratio Test, don’t try the Root Test because L will again be 1. And if L ෇ 1 in the Root Test, don’t try the Ratio Test because it will fail too.)

ϱ 2n ϩ 3 n v EXAMPLE 6 Test the convergence of the series ͚ ͩ ͪ . n෇1 3n ϩ 2 SOLUTION 2n ϩ 3 n a ෇ ͩ ͪ n 3n ϩ 2 ϩ 3 ϩ 2 n 2n 3 n 2 sԽ an Խ ෇ ෇ l Ͻ 1 3n ϩ 2 2 3 3 ϩ n

Thus the given series converges by the Root Test.

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SECTION 11.6 ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS 737

Rearrangements The question of whether a given convergent series is absolutely convergent or conditionally convergent has a bearing on the question of whether infinite sums behave like finite sums. If we rearrange the order of the terms in a finite sum, then of course the value of the sum remains unchanged. But this is not always the case for an infinite series. By a rearrangement of an infinite series͸ a we mean a series obtained by simply changing the order of n ͸ the terms. For instance, a rearrangement of an could start as follows:

a1 ϩ a2 ϩ a5 ϩ a3 ϩ a4 ϩ a15 ϩ a6 ϩ a7 ϩ a20 ϩиии It turns out that if ͸ a is an absolutely convergent series with sum s, n ͸ then any rearrangement of an has the same sum s. However, any conditionally convergent series can be rearranged to give a different sum. To illustrate this fact let’s consider the alternating harmonic series 6 Ϫ 1 ϩ 1 Ϫ 1 ϩ 1 Ϫ 1 ϩ 1 Ϫ 1 ϩиии෇ 1 2 3 4 5 6 7 8 ln 2 1 (See Exercise 36 in Section 11.5.) If we multiply this series by 2 , we get 1 Ϫ 1 ϩ 1 Ϫ 1 ϩиии෇ 1 2 4 6 8 2 ln 2 Inserting zeros between the terms of this series, we have Adding these zeros does not affect the sum of the series; each term in the sequence of partial 7 ϩ 1 ϩ Ϫ 1 ϩ ϩ 1 ϩ Ϫ 1 ϩиии෇ 1 0 2 0 4 0 6 0 8 2 ln 2 sums is repeated, but the limit is the same. Now we add the series in Equations 6 and 7 using Theorem 11.2.8: 8 ϩ 1 Ϫ 1 ϩ 1 ϩ 1 Ϫ 1 ϩиии෇ 3 1 3 2 5 7 4 2 ln 2 Notice that the series in 8 contains the same terms as in 6 , but rearranged so that one negative term occurs after each pair of positive terms. The sums of these series, however, are different. In fact, Riemann proved that if ͸ a is a conditionally convergent series and r is any real number what- n ͸ soever, then there is a rearrangement of an that has a sum equal to r. A proof of this fact is outlined in Exercise 44.

11.6 Exercises

͸ ϱ n ϱ n 1. What can you say about the series an in each of the following ͑Ϫ1͒ ͑Ϫ3͒ 5. ͚ 6. ͚ cases? n෇0 5n ϩ 1 n෇0 ͑2n ϩ 1͒!

anϩ1 anϩ1 (a) lim Ϳ Ϳ ෇ 8 (b) lim Ϳ Ϳ ෇ 0.8 ϱ ϱ l ϱ l ϱ n! n an n an 7. 2 k 8. ͚ k( 3 ) ͚ n k෇1 n෇1 100 a ϩ (c) lim Ϳ n 1 Ϳ ෇ 1 l ϱ ϱ ϱ n an ͑1.1͒n n 9. ͑Ϫ ͒ n 10. ͑Ϫ ͒ n ͚ 1 4 ͚ 1 ෇ n ෇ s 3 ϩ 2–30 Determine whether the series is absolutely convergent, n 1 n 1 n 2 conditionally convergent, or divergent. ϱ ͑Ϫ1͒n e 1͞n ϱ sin 4n ϱ n 11. 12. ͑Ϫ2͒ ͚ 3 ͚ n 2. n෇1 n n෇1 4 ͚ 2 n෇1 n ϱ n ϱ n ϱ 10n ϱ n10 3. 4. ͑Ϫ ͒nϪ1 13. ͚ 14. ͚ ͚ n ͚ 1 2 2nϩ1 nϩ1 n෇1 5 n෇1 n ϩ 4 n෇1 ͑n ϩ 1͒4 n෇1 ͑Ϫ10͒

1. Homework Hints available at stewartcalculus.com

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738 CHAPTER 11 INFINITE SEQUENCES AND SERIES

ϱ ͑Ϫ1͒n arctan n ϱ 3 Ϫ cos n 36. For which positive integers k is the following series convergent? 15. 16. ͚ 2 ͚ 2͞3 ෇ n ෇ n Ϫ 2 ϱ 2 n 1 n 1 ͚ ͑n!͒ ϱ n ϱ ͑Ϫ1͒ n! n෇1 ͑kn͒! 17. 18. ͚ ͚ n n෇2 ln n n෇1 n ͸ϱ n 37. (a) Show that n෇0 x ͞n! converges for all x . ϱ ϱ n n cos͑n␲͞3͒ ͑Ϫ2͒ (b) Deduce that limn l ϱ x ͞n! ෇ 0 for all x . 19. 20. ͚ ͚ n n෇1 n! n෇1 n ͸ 38. Letan be a series with positive terms and let rn ෇ anϩ1͞an. ϱ n ϱ 5n ෇ Ͻ ͸ n2 ϩ 1 Ϫ2n Suppose that limn l ϱ rn L 1, so an converges by the 21. ͩ ͪ 22. ͚ ͩ ͪ ͚ 2 Ratio Test. As usual, we let Rn be the remainder after n terms, ෇ 2n ϩ 1 ෇ n ϩ 1 n 1 n 2 that is, 2 ϱ 1 n ϱ ͑2n͒! ෇ ϩ ϩ ϩиии 23. ͩ ϩ ͪ 24. ͚ Rn anϩ1 anϩ2 anϩ3 ͚ 1 2 n෇1 n n෇1 ͑n!͒ (a) If ͕rn ͖ is a decreasing sequence and rnϩ1 Ͻ 1 , show, by ϱ n100100n ϱ 2n 2 25. ͚ 26. ͚ summing a geometric series, that n෇1 n! n෇1 n! anϩ1 Rn ഛ 7 ؒ 5 ؒ 3 ؒ 1 5 ؒ 3 ؒ 1 3 ؒ 1 27. 1 Ϫ ϩ Ϫ ϩиии 1 Ϫ rnϩ1 3! 5! 7! иии ؒ ͑2n Ϫ 1͒ (b) If ͕rn ͖ is an increasing sequence, show that ؒ 5 ؒ 3 ؒ 1 ϩ ͑Ϫ1͒nϪ1 ϩиии ͑2n Ϫ 1͒! a ϩ R ഛ n 1 n Ϫ 14 ؒ 10 ؒ 6 ؒ 2 10 ؒ 6 ؒ 2 6 ؒ 2 2 28. ϩ ϩ ϩ ϩиии 1 L 14 ؒ 11 ؒ 8 ؒ 5 11 ؒ 8 ؒ 5 8 ؒ 5 5 ͸ϱ n 39. (a) Find the partial sum s5 of the series n෇1 1͑͞n2 ͒. Use Exer- ϱ 2 ؒ 4 ؒ 6 ؒ иии ؒ ͑2n͒ 29. ͚ cise 38 to estimate the error in using s5 as an approximation n෇1 n! to the sum of the series. (b) Find a value of ns so that is within 0.00005 of the sum. ϱ 2n n! n 30. ͚ ͑Ϫ1͒n Use this value of n to approximate the sum of the series. n෇1 5 ؒ 8 ؒ 11 ؒ иии ؒ ͑3n ϩ 2͒ 40. Use the sum of the first 10 terms to approximate the sum of the series 31. The terms of a series are defined recursively by the equations ϱ n ͚ n ϩ n෇1 2 ෇ ෇ 5n 1 a1 2 anϩ1 ϩ an 4n 3 Use Exercise 38 to estimate the error. ͸ Determine whether an converges or diverges. 41. Prove the Root Test. [Hint for part (i): Take any number r such ͸ that L Ͻ r Ͻ 1 and use the fact that there is an integer N such 32. A series an is defined by the equations n that sԽ an Խ Ͻ rn whenever ജ N.] 2 ϩ cos n 42. Around 1910, the Indian mathematician Srinivasa Ramanujan a1 ෇ 1 anϩ1 ෇ an sn discovered the formula

͸ ϱ Determine whether an converges or diverges. s ͑ ͒ ͑ ϩ ͒ 1 ෇ 2 2 4n ! 1103 26390n ͚ 4 4n ␲ 9801 n෇0 ͑n!͒ 396 33–34 Let ͕bn͖ be a sequence of positive numbers that converges 1 to 2 . Determine whether the given series is absolutely convergent. William Gosper used this series in 1985 to compute the first 17 million digits of ␲ . ϱ b n cos n␲ ϱ ͑Ϫ1͒n n! 33. ͚ n 34. ͚ (a) Verify that the series is convergent. n иии n෇1 n n෇1 n b1b2 b3 bn (b) How many correct decimal places of ␲ do you get if you use just the first term of the series? What if you use two 35. For which of the following series is the Ratio Test inconclusive terms? ͸ ͸ ϩ (that is, it fails to give a definite answer)? 43. Given any series an , we define a series an whose terms are ϱ ϱ ͸ ͸ Ϫ 1 n all the positive terms of an and a series an whose terms (a) ͚ (b) ͚ ͸ 3 n are all the negative terms of an . To be specific, we let n෇1 n n෇1 2 ϱ ͑Ϫ ͒nϪ1 ϱ s 3 n an ϩ Խ an Խ an Ϫ Խ an Խ ϩ ෇ Ϫ ෇ (c) ͚ (d) ͚ 2 an an n෇1 sn n෇1 1 ϩ n 2 2

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SECTION 11.7 STRATEGY FOR TESTING SERIES 739

ϩ Ϫ ϩ Notice that if an Ͼ 0 , then an ෇ an and an ෇ 0 , whereas if Take just enough positive terms an so that their sum is greater Ϫ ϩ Ϫ an Ͻ 0, then an ෇ an and an ෇ 0 . than ra . Then add just enough negative terms n so that the ͸ (a) If an is absolutely convergent, show that both of the cumulative sum is less than r . Continue in this manner and use series ͸ aϩ and ͸ aϪ are convergent. Theorem 11.2.6.] ͸ n n (b) If an is conditionally convergent, show that both of the ͸ 45. Suppose the series an is conditionally convergent. ͸ ϩ ͸ Ϫ series an and an are divergent. ͸ 2 (a) Prove that the series n an is divergent. (b) Conditional convergence of͸ a is not enough to deter- ͸ ͸ n 44. Prove that if an is a conditionally convergent series and mine whethernan is convergent. Show this by giving an r is any real number, then there is a rearrangement of ͸ a example of a conditionally convergent series such that n ͸ ͸ whose sum is r . [Hints: Use the notation of Exercise 43. nan converges and an example where nan diverges.

11.7 Strategy for Testing Series

We now have several ways of testing a series for convergence or divergence; the problem is to decide which test to use on which series. In this respect, testing series is similar to integrating functions. Again there are no hard and fast rules about which test to apply to a given series, but you may find the following advice of some use. It is not wise to apply a list of the tests in a specific order until one finally works. That would be a waste of time and effort. Instead, as with integration, the main strategy is to classify the series according to its form. 1. If the series is of the form ͸ 1͞n p , it is a p -series, which we know to be convergent if p Ͼ 1 and divergent if p ഛ 1 . 2. If the series has the form ͸ ar nϪ1 or ͸ ar n , it is a geometric series, which converges if Խ r Խ Ͻ 1 and diverges if Խ r Խ ജ 1 . Some preliminary algebraic manipulation may be required to bring the series into this form. 3. If the series has a form that is similar to a p -series or a geometric series, then

one of the comparison tests should be considered. In particular, if an is a rational function or an algebraic function of n (involving roots of polynomials), then the series should be compared with a p -series. Notice that most of the series in Exer- cises 11.4 have this form. (The value of p should be chosen as in Sec tion11.4 by keeping only the highest powers of n in the numerator and denominator.) The comparison tests apply only to series with positive terms, but if ͸ a has some negative ͸ n terms, then we can apply the Comparison Test to Խ an Խ and test for absolute convergence.

4. If you can see at a glance that lim n l ϱ an 0, then the Test for Divergence should be used. ͸ nϪ1 ͸ n 5. If the series is of the form ͑Ϫ1͒ bn or ͑Ϫ1͒ bn , then the Alternating Series Test is an obvious possibility. 6. Series that involve factorials or other products (including a constant raised to the nth power) are often conveniently tested using the Ratio Test. Bear in mind that

Խ anϩ1͞an Խl 1 as n l ϱ for all p -series and therefore all rational or algebraic functions of n . Thus the Ratio Test should not be used for such series.

n 7. If an is of the form ͑bn ͒ , then the Root Test may be useful. 8. ෇ ͑ ͒ xϱ ͑ ͒ If an f n , where 1 f x dx is easily evaluated, then the Integral Test is effective (assuming the hypotheses of this test are satisﬁed).

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740 CHAPTER 11 INFINITE SEQUENCES AND SERIES

In the following examples we don’t work out all the details but simply indicate which tests should be used.

ϱ n Ϫ 1 v EXAMPLE 1 ͚ n෇1 2n ϩ 1 l 1 l ϱ Since an 2 0 as n , we should use the Test for Divergence.

ϱ sn3 ϩ 1 EXAMPLE 2 ͚ 3 2 n෇1 3n ϩ 4n ϩ 2 Since a is an algebraic function of np , we compare the given series with a -series. The n ͸ comparison series for the Limit Comparison Test is bn , where

sn3 n3͞2 1 b ෇ ෇ ෇ n 3n3 3n3 3n3͞2

ϱ v EXAMPLE 3 ͚ neϪn2 n෇1 xϱ Ϫx2 Since the integral 1 xe dx is easily evaluated, we use the Integral Test. The Ratio Test also works.

ϱ n3 EXAMPLE 4 ͑Ϫ ͒n ͚ 1 4 n෇1 n ϩ 1 Since the series is alternating, we use the Alternating Series Test.

ϱ 2k v EXAMPLE 5 ͚ k෇1 k! Since the series involves k! , we use the Ratio Test.

ϱ 1 EXAMPLE 6 ͚ n n෇1 2 ϩ 3 Since the series is closely related to the geometric series ͸ 1͞3n , we use the Comparison Test.

11.7 Exercises

1–38 ϱ ϱ Test the series for convergence or divergence. 1 1 1 ϱ ϱ 11. ͩ ϩ ͪ 12. ͚ n ͚ 1 ͑2n ϩ 1͒ 3 n 2 ϩ 1. 2. n෇1 n 3 k෇1 ksk 1 ͚ n ͚ 2n n෇1 n ϩ 3 n෇1 n ϱ 3n n2 ϱ sin 2n ϱ ϱ 13. ͚ 14. ͚ n n n n n 3. ͑Ϫ ͒ 4. ͑Ϫ ͒ n෇1 n! n෇1 1 ϩ 2 ͚ 1 ͚ 1 2 n෇1 n ϩ 2 n෇1 n ϩ 2 ϱ kϪ1 kϩ1 ϱ 2 ϱ Ϫ ϱ 2 3 n ϩ 1 n2 2n 1 1 15. ͚ 16. ͚ 5. 6. k 3 ͚ n ͚ k෇1 k n෇1 n ϩ 1 n෇1 ͑Ϫ5͒ n෇1 2n ϩ 1 ϱ ͒ ϱ ϱ ؒ ؒ ؒ иииؒ ͑ Ϫ 1 2 k k! 1 3 5 2n 1 7. ͚ 8. ͚ 17. ͚ n෇1 2 ؒ 5 ؒ 8 ؒ иии ؒ ͑3n Ϫ 1͒ n෇2 nsln n k෇1 ͑k ϩ 2͒! ϱ ϱ ϱ ͑Ϫ1͒nϪ1 9. ͚ k 2eϪk 10. ͚ n2eϪn 3 18. ͚ k෇1 n෇1 n෇2 sn Ϫ 1

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SECTION 11.8 POWER SERIES 741

ϱ ln n ϱ s3 k Ϫ 1 ϱ ͑Ϫ1͒n ϱ sj 19. ͚ ͑Ϫ1͒n 20. ͚ 29. ͚ 30. ͚ ͑Ϫ1͒ j n෇1 sn k෇1 k(sk ϩ 1) n෇1 cosh n j෇1 j ϩ 5

ϱ ϱ ϱ ϱ 1 5 k ͑n!͒n 21. ͑Ϫ ͒n ͑ ͞ 2͒ 22. 31. ͚ 32. ͚ ͚ 1 cos 1 n ͚ k ϩ k 4n n෇1 k෇1 2 ϩ sin k k෇1 3 4 n෇1 n

2 ϱ ϱ ϱ n n ϱ 1 23. ͑ ͞ ͒ 24. ͑ ͞ ͒ 33. ͚ ͩ ͪ 34. ͚ ͚ tan 1 n ͚ n sin 1 n ϩ ϩ 2 n෇1 n෇1 n෇1 n 1 n෇1 n n cos n ϱ ϱ ϱ ϱ 2 1 1 n! n ϩ 1 35. ͚ 36. ͚ 25. ͚ 26. ͚ 1ϩ1͞n ln n n 2 n n෇1 n n෇2 ͑ln n͒ n෇1 e n෇1 5 ϱ ϱ n ϱ k ln k ϱ e 1͞n 37. ͚ (sn 2 Ϫ 1) 38. ͚ (sn 2 Ϫ 1) 27. 28. ෇ ෇ ͚ 3 ͚ 2 n 1 n 1 k෇1 ͑k ϩ 1͒ n෇1 n

11.8 Power Series

A power series is a series of the form

ϱ n 2 3 1 ͚ cn x ෇ c0 ϩ c1 x ϩ c2 x ϩ c3 x ϩиии n෇0

where xc is a variable and the n ’s are constants called the coefﬁcients of the series. For each ﬁxed x , the series 1 is a series of constants that we can test for convergence or divergence. A power series may converge for some values of xx and diverge for other values of . The sum of the series is a function

2 n f ͑x͒ ෇ c0 ϩ c1 x ϩ c2 x ϩиииϩcn x ϩиии

whose domain is the set of all xf for which the series converges. Notice that resembles a polynomial. The only difference is that f has inﬁnitely many terms.

Trigonometric Series For instance, if we take cn ෇ 1 for all n , the power series becomes the geometric series A power series is a series in which each term is ϱ a power function. A trigonometric series ͚ x n ෇ 1 ϩ x ϩ x 2 ϩиииϩx n ϩиии ϱ n෇0

͚ ͑an cos nx ϩ bn sin nx͒ n෇0 which converges when Ϫ1 Ͻ x Ͻ 1 and diverges when Խ x Խ ജ 1 . (See Equation 11.2.5.) is a series whose terms are trigonometric More generally, a series of the form functions. This type of series is discussed on ϱ the website n 2 2 ͚ cn͑x Ϫ a͒ ෇ c0 ϩ c1͑x Ϫ a͒ ϩ c2͑x Ϫ a͒ ϩиии www.stewartcalculus.com n෇0 Click on Additional Topics and then on Fourier is called a power series in ͑x Ϫ a͒ or a power series centered at a or a power series about Series. a. Notice that in writing out the term corresponding to n ෇ 0 in Equations 1 and 2 we have adopted the convention that ͑x Ϫ a͒0 ෇ 1 even when x ෇ ax . Notice also that when ෇ a all of the terms are 0 for n ജ 1 and so the power series 2 always converges when x ෇ a .

ϱ v EXAMPLE 1 For what values of x is the series ͚ n!x n convergent? n෇0

SOLUTION We use the Ratio Test. If we let an , as usual, denote the nth term of the series, n then an ෇ n!x . If x 0 , we have Notice that nϩ1 n ϩ 1͒! ෇ ͑n ϩ 1͒n͑n Ϫ 1͒ ؒ . . . ؒ 3 ؒ 2 ؒ 1 anϩ1 ͑n ϩ 1͒!x͑ lim Ϳ Ϳ ෇ lim Ϳ Ϳ ෇ lim ͑n ϩ 1͒Խ x Խ ෇ ϱ l ϱ l ϱ n l ϱ ෇ ͑n ϩ 1͒n! n an n n!x n

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742 CHAPTER 11 INFINITE SEQUENCES AND SERIES

By the Ratio Test, the series diverges when x 0 . Thus the given series converges only when x ෇ 0.

ϱ ͑x Ϫ 3͒n v EXAMPLE 2 For what values of x does the series ͚ converge? n෇1 n

n SOLUTION Let an ෇ ͑x Ϫ 3͒ ͞n. Then

͑ Ϫ ͒nϩ1 Ϳ anϩ1 Ϳ ෇ Ϳ x 3 ؒ n Ϳ n an n ϩ 1 ͑x Ϫ 3͒ 1 ෇ Խ x Ϫ 3 Խ l Խ x Ϫ 3 Խ as n l ϱ 1 1 ϩ n

By the Ratio Test, the given series is absolutely convergent, and therefore convergent, when Խ x Ϫ 3 Խ Ͻ 1 and divergent when Խ x Ϫ 3 Խ Ͼ 1. Now

Խ x Ϫ 3 Խ Ͻ 1 &? Ϫ1 Ͻ x Ϫ 3 Ͻ 1 &? 2 Ͻ x Ͻ 4

so the series converges when 2 Ͻ x Ͻ 4 and diverges when x Ͻ 2 or x Ͼ 4 . The Ratio Test gives no information when Խ x Ϫ 3 Խ ෇ 1 so we must consider x ෇ 2 and x ෇ 4 separately. If we put x ෇ 4 in the series, it becomes ͸ 1͞n , the harmonic series, which is divergent. If x ෇ 2 , the series is ͸ ͑Ϫ1͒n͞n , which converges by the Alternating Series Test. Thus the given power series converges for 2 ഛ x Ͻ 4 .

We will see that the main use of a power series is that it provides a way to represent some of the most important functions that arise in mathematics, physics, and chemistry. In particular, the sum of the power series in the next example is called a Bessel function, after the German astronomer Friedrich Bessel (1784Ð1846), and the function given in Exercise35 is another example of a Bessel function. In fact, these functions ﬁrst arose when Bessel solved Kepler’s equation for describing planetary motion. Since that time, these functions have been applied in many different physical situations, including the temperature distri- National Film Board of Canada bution in a circular plate and the shape of a vibrating drumhead.

EXAMPLE 3 Find the domain of the Bessel function of order 0 deﬁned by

ϱ ͑Ϫ ͒n 2n ͑ ͒ ෇ 1 x J0 x ͚ 2n 2 n෇0 2 ͑n!͒

n 2n 2n 2 SOLUTION Let an ෇ ͑Ϫ1͒ x ͓͞2 ͑n!͒ ͔. Then

nϩ1 2͑nϩ1͒ 2n 2 anϩ1 ͑Ϫ1͒ x 2 ͑n!͒ Notice how closely the computer-generated Ϳ Ϳ ෇ Ϳ ؒ Ϳ 2͑nϩ1͓͒͑ ϩ ͒ ͔2 ͑Ϫ ͒n 2n model (which involves Bessel functions and an 2 n 1 ! 1 x cosine functions) matches the photograph of a 2nϩ2 2n͑ ͒2 !vibrating rubber membrane. ෇ x ؒ 2 n 22nϩ2͑n ϩ 1͒2͑n!͒2 x 2n x 2 ෇ l 0 Ͻ 1 for all x 4͑n ϩ 1͒2

Thus, by the Ratio Test, the given series converges for all values of x . In other words,

the domain of the Bessel function J0 is ͑Ϫϱ, ϱ͒ ෇ ޒ.

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SECTION 11.8 POWER SERIES 743

y Recall that the sum of a series is equal to the limit of the sequence of partial sums. So s™ when we deﬁne the Bessel function in Example 3 as the sum of a series we mean that, for 1 s¸ every real number x , n ͑Ϫ1͒ix 2i J0͑x͒ ෇ lim sn͑x͒ where sn͑x͒ ෇ ͚ l ϱ 2i 2 s¢ n i෇0 2 ͑i!͒

0 x 1 The first few partial sums are J¸ s¡ s£ x 2 x 2 x 4 s0͑x͒ ෇ 1 s1͑x͒ ෇ 1 Ϫ s2͑x͒ ෇ 1 Ϫ ϩ FIGURE 1 4 4 64 Partial sums of the Bessel function J¸ x 2 x 4 x 6 x 2 x 4 x 6 x 8 s ͑x͒ ෇ 1 Ϫ ϩ Ϫ s ͑x͒ ෇ 1 Ϫ ϩ Ϫ ϩ y 3 4 64 2304 4 4 64 2304 147,456 1 y=J¸(x) Figure 1 shows the graphs of these partial sums, which are polynomials. They are all approximations to the function J0 , but notice that the approximations become better when more terms are included. Figure 2 shows a more complete graph of the Bessel function. _10 10 For the power series that we have looked at so far, the set of values of x for which the 0 x series is convergent has always turned out to be an interval [a finite interval for the geometric series and the series in Example 2, the infinite interval ͑Ϫϱ, ϱ͒ in Example 3, and a col- lapsed interval ͓0, 0͔ ෇ ͕0͖ in Example 1]. The following theorem, proved in Appendix F, FIGURE 2 says that this is true in general.

ϱ n 3 Theorem For a given power series ͚ cn͑x Ϫ a͒ there are only three possibilities: n෇0 (i) The series converges only when x ෇ a . (ii) The series converges for all x . (iii) There is a positive number R such that the series converges if Խ x Ϫ a Խ Ͻ R and diverges if Խ x Ϫ a Խ Ͼ R.

The number R in case (iii) is called the radius of convergence of the power series. By convention, the radius of convergence is R ෇ 0 in case (i) and R ෇ ϱ in case (ii). The interval of convergence of a power series is the interval that consists of all values of x for which the series converges. In case (i) the interval consists of just a single point a . In case (ii) the interval is ͑Ϫϱ, ϱ͒ . In case (iii) note that the inequality Խ x Ϫ a Խ Ͻ R can be rewritten as a Ϫ R Ͻ x Ͻ a ϩ R. When x is an endpoint of the interval, that is, x ෇ a Ϯ R , anything can happen—the series might converge at one or both endpoints or it might diverge at both endpoints. Thus in case (iii) there are four possibilities for the interval of convergence:

͑a Ϫ R, a ϩ R͒͑a Ϫ R, a ϩ R͔͓a Ϫ R, a ϩ R͓͒a Ϫ R, a ϩ R͔

The situation is illustrated in Figure 3.

convergence for |x-a|

a-R a a+R

FIGURE 3 divergence for |x-a|>R

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744 CHAPTER 11 INFINITE SEQUENCES AND SERIES

We summarize here the radius and interval of convergence for each of the examples already considered in this section.

Series Radius of convergence Interval of convergence

ϱ Geometric series ͚ x n R ෇ 1 ͑Ϫ1, 1͒ n෇0

ϱ Example 1 ͚ n! x n R ෇ 0 ͕0͖ n෇0

ϱ ͑x Ϫ 3͒n Example 2 ͚ R ෇ 1 ͓2, 4͒ n෇1 n

ϱ ͑Ϫ ͒n 2n 1 x ෇ ϱ ͑Ϫϱ ϱ͒ Example 3 ͚ 2n 2 R , n෇0 2 ͑n!͒

In general, the Ratio Test (or sometimes the Root Test) should be used to determine the radius of convergence R . The Ratio and Root Tests always fail when x is an endpoint of the interval of convergence, so the endpoints must be checked with some other test.

EXAMPLE 4 Find the radius of convergence and interval of convergence of the series

ϱ ͚ ͑Ϫ3͒nx n n෇0 sn ϩ 1

n n SOLUTION Let an ෇ ͑Ϫ3͒ x ͞sn ϩ 1. Then

nϩ1 nϩ1 a ϩ ͑Ϫ3͒ x sn ϩ 1 n ϩ 1 Ϳ n 1 Ϳ ෇ Ϳ ؒ Ϳ ෇ Ϳ Ϫ3xͱ Ϳ n n an sn ϩ 2 ͑Ϫ3͒ x n ϩ 2

1 ϩ ͑1͞n͒ ෇ 3ͱ Խ x Խ l 3 Խ x Խ as n l ϱ 1 ϩ ͑2͞n͒

By the Ratio Test, the given series converges if 3 Խ x Խ Ͻ 1 and diverges if 3 Խ x Խ Ͼ 1 . Ͻ 1 Ͼ 1 Thus it converges if Խ x Խ 3 and diverges if Խ x Խ 3 . This means that the radius of con- ෇ 1 vergence is R 3 . Ϫ 1 1 We know the series converges in the interval ( 3, 3 ) , but we must now test for con- ෇ Ϫ 1 vergence at the endpoints of this interval. If x 3 , the series becomes

ϱ ͑Ϫ3͒n (Ϫ1 )n ϱ 1 1 1 1 1 ͚ 3 ෇ ͚ ෇ ϩ ϩ ϩ ϩиии n෇0 sn ϩ 1 n෇0 sn ϩ 1 s1 s2 s3 s4

which diverges. (Use the Integral Test or simply observe that it is a p -series with ෇ 1 Ͻ ෇ 1 p 2 1.) If x 3 , the series is

ϱ ͑Ϫ3͒n ( 1 )n ϱ ͑Ϫ1͒n ͚ 3 ෇ ͚ n෇0 sn ϩ 1 n෇0 sn ϩ 1

which converges by the Alternating Series Test. Therefore the given power series con- Ϫ1 Ͻ ഛ 1 Ϫ1 1 verges when 3 x 3 , so the interval of convergence is ( 3, 3 ] .

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SECTION 11.8 POWER SERIES 745

v EXAMPLE 5 Find the radius of convergence and interval of convergence of the series

ϱ n͑x ϩ 2͒n ͚ nϩ1 n෇0 3

n nϩ1 SOLUTION If an ෇ n͑x ϩ 2͒ ͞3 , then

͑ ϩ ͒͑ ϩ ͒nϩ1 nϩ1 Ϳ anϩ1 Ϳ ෇ Ϳ n 1 x 2 ؒ 3 Ϳ nϩ2 n an 3 n͑x ϩ 2͒

1 Խ x ϩ 2 Խ Խ x ϩ 2 Խ ෇ ͩ1 ϩ ͪ l as n l ϱ n 3 3

Using the Ratio Test, we see that the series converges if Խ x ϩ 2 Խ͞3 Ͻ 1 and it diverges if Խ x ϩ 2 Խ͞3 Ͼ 1. So it converges if Խ x ϩ 2 Խ Ͻ 3 and diverges if Խ x ϩ 2 Խ Ͼ 3. Thus the radius of convergence is R ෇ 3 . The inequality Խ x ϩ 2 Խ Ͻ 3 can be written as Ϫ5 Ͻ x Ͻ 1, so we test the series at the endpoints Ϫ5 and 1. When x ෇ Ϫ5 , the series is

ϱ n͑Ϫ3͒n ϱ ෇ 1 ͑Ϫ ͒n ͚ nϩ1 3 ͚ 1 n n෇0 3 n෇0

which diverges by the Test for Divergence [͑Ϫ1͒nnx doesn’t converge to 0]. When ෇ 1 , the series is

ϱ n͑ ͒n ϱ 3 ෇ 1 ͚ nϩ1 3 ͚ n n෇0 3 n෇0

which also diverges by the Test for Divergence. Thus the series converges only when Ϫ5 Ͻ x Ͻ 1, so the interval of convergence is ͑Ϫ5, 1͒ .

11.8 Exercises

1. What is a power series? ϱ n 2 x n ϱ 10 nx n 9. ͑Ϫ ͒n 10. ͚ 1 n ͚ 3 2. (a) What is the radius of convergence of a power series? n෇1 2 n෇1 n How do you ﬁnd it? ϱ ͑Ϫ3͒n ϱ x n (b) What is the interval of convergence of a power series? 11. ͚ x n 12. ͚ s n How do you ﬁnd it? n෇1 n n n෇1 n3 ϱ ϱ ϩ 3–28 x n x 2n 1 Find the radius of convergence and interval of convergence 13. ͚ ͑Ϫ1͒n 14. ͚ ͑Ϫ1͒n n ͑ ϩ ͒ of the series. n෇2 4 ln n n෇0 2n 1 ! ϱ ϱ ͑Ϫ1͒nx n ϱ ͑x Ϫ 2͒n ϱ ͑x Ϫ 3͒ n 3. ͚ ͑Ϫ1͒nnxn 4. ͚ 15. ͚ 16. ͚ ͑Ϫ1͒n 3 2 n෇1 n෇1 sn n෇0 n ϩ 1 n෇0 2n ϩ 1 ϱ x n ϱ ͑Ϫ1͒nx n ϱ 3n͑x ϩ 4͒n ϱ n 5. 6. 17. 18. ͑ ϩ ͒n ͚ ͚ 2 ͚ ͚ n x 1 n෇1 2n Ϫ 1 n෇1 n n෇1 sn n෇1 4 ϱ x n ϱ ϱ ͑x Ϫ 2͒n ϱ ͑2x Ϫ 1͒n 7. ͚ 8. ͚ n nx n 19. 20. ͚ n ͚ n n෇0 n! n෇1 n෇1 n n෇1 5 sn

; Graphing calculator or computer required CAS Computer algebra system required 1. Homework Hints available at stewartcalculus.com

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746 CHAPTER 11 INFINITE SEQUENCES AND SERIES

ϱ n 35. The function J defined by 21. ͑ Ϫ ͒n Ͼ 1 ͚ n x a , b 0 n෇1 b ϱ ͑Ϫ1͒nx 2nϩ1 ϱ n ͑ ͒ ෇ J1 x ͚ ϩ b n ͑ ϩ ͒ 2n 1 22. ͚ ͑x Ϫ a͒ , b Ͼ 0 n෇0 n! n 1 !2 n෇2 ln n ϱ ϱ 2 n is called the Bessel function of order 1. n n x 23. ͚ n!͑2x Ϫ 1͒ 24. ͚ (a) Find its domain. n෇1 n෇1 2 ؒ 4 ؒ 6 ؒ иии ؒ ͑2n͒ ; (b) Graph the first several partial sums on a common ϱ ͑5x Ϫ 4͒n ϱ x 2n 25. 26. screen. ͚ 3 ͚ 2 n෇1 n n෇2 n͑ln n͒ CAS (c) If your CAS has built-in Bessel functions, graph J1 on the ϱ same screen as the partial sums in part (b) and observe x n 27. ͚ how the partial sums approximate J1 . n෇1 1 ؒ 3 ؒ 5 ؒ иии ؒ ͑2n Ϫ 1͒ 36. The function A defined by ϱ n!x n 28. ͚ 3 6 9 ෇ ؒ ؒ ؒ иии ؒ ͑ n Ϫ ͒ x x x n 1 1 3 5 2 1 A͑x͒ ෇ 1 ϩ ϩ ϩ ϩиии 2 и 3 2 и 3 и 5 и 6 2 и 3 и 5 и 6 и 8 и 9

͸ϱ n 29. If n෇0 cn 4 is convergent, does it follow that the following is called an Airy function after the English mathematician series are convergent? and astronomer Sir George Airy (1801Ð1892). ϱ ϱ (a) Find the domain of the Airy function. n n (a) ͚ cn͑Ϫ2͒ (b) ͚ cn͑Ϫ4͒ ; (b) Graph the first several partial sums on a common screen. n෇0 n෇0 CAS (c) If your CAS has built-in Airy functions, graph A on the ͸ϱ n 30. Suppose that n෇0 cn x converges when x ෇ Ϫ4 and diverges same screen as the partial sums in part (b) and observe when x ෇ 6 . What can be said about the convergence or how the partial sums approximate A . divergence of the following series? 37. A function f is defined by ϱ ϱ n (a) ͚ cn (b) ͚ cn 8 f ͑x͒ ෇ 1 ϩ 2x ϩ x 2 ϩ 2x 3 ϩ x 4 ϩиии n෇0 n෇0 ϱ ϱ that is, its coefficients are c2n ෇ 1 and c2nϩ1 ෇ 2 for all ͑Ϫ ͒n ͑Ϫ ͒n n (c) ͚ cn 3 (d) ͚ 1 cn 9 n ജ 0. Find the interval of convergence of the series and find n෇0 n෇0 an explicit formula for f ͑x͒ . 31. If k is a positive integer, find the radius of convergence of 38. ͑ ͒ ෇ ͸ϱ n ෇ ജ the series If f x n෇0 cn x , where cnϩ4 cn for all n 0 , find the ϱ ͑ ͒k interval of convergence of the series and a formula for f ͑x͒ . n! n ͚ x n n෇0 ͑kn͒! 39. Show that if lim n l ϱ sԽ cn Խ ෇ cc, where 0 , then the radius of convergence of the power series ͸ c x n is R ෇ 1͞c . 32. Let and qp be real numbers with p Ͻ q . Find a power series n ͸ n whose interval of convergence is 40. Suppose that the power series cn͑x Ϫ a͒ satisfies cn 0 (a) ͑p, q͒ (b) ͑p, q͔ for all n . Show that if lim n l ϱ Խ cn͞cnϩ1 Խ exists, then it is equal (c) ͓p, q͒ (d) ͓p, q͔ to the radius of convergence of the power series. ͸ n 33. Is it possible to find a power series whose interval of conver- 41. Suppose the series cn x has radius of convergence 2 and ͸ n gence is ͓0, ϱ͒ ? Explain. the series dn x has radius of convergence 3. What is the ͸ ͑ ϩ ͒ n ͸ϱ n radius of convergence of the series cn dn x ? ; 34. Graph the first several partial sums sn͑x͒ of the series n෇0 x , together with the sum function f ͑x͒ ෇ 1͑͞1 Ϫ x͒, on a com- 42. Suppose that the radius of convergence of the power series ͸ n mon screen. On what interval do these partial sums appear to cn x is R . What is the radius of convergence of the power ͸ 2n be converging to f ͑x͒ ? series cn x ?

11.9 Representations of Functions as Power Series

In this section we learn how to represent certain types of functions as sums of power series by manipulating geometric series or by differentiating or integrating such a series. You might wonder why we would ever want to express a known function as a sum of inﬁnitely many terms. We will see later that this strategy is useful for integrating functions that don’t have elementary antiderivatives, for solving differential equations, and for approximating func-

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SECTION 11.9 REPRESENTATIONS OF FUNCTIONS AS POWER SERIES 747

tions by polynomials. (Scientists do this to simplify the expressions they deal with; computer scientists do this to represent functions on calculators and computers.) We start with an equation that we have seen before:

1 ϱ 1 ෇ 1 ϩ x ϩ x 2 ϩ x 3 ϩиии෇ ͚ x n Խ x Խ Ͻ 1 1 Ϫ x n෇0

A geometric illustration of Equation 1 is shown We ﬁrst encountered this equation in Example 6 in Section 11.2, where we obtained it by in Figure 1. Because the sum of a series is the observing that the series is a geometric series with a ෇ 1 and r ෇ x . But here our point of limit of the sequence of partial sums, we have view is different. We now regard Equation 1 as expressing the function f ͑x͒ ෇ 1͑͞1 Ϫ x͒ as a sum of a power series. 1 ෇ ͑ ͒ lim sn x s¡¡ 1 Ϫ x n l ϱ y where sˆ s∞ ͑ ͒ ෇ ϩ ϩ 2 ϩиииϩ n sn x 1 x x x f is the n th partial sum. Notice that as n

increases, sn͑x͒ becomes a better approximation to f ͑x͒ for Ϫ1 Ͻ x Ͻ 1. s™

FIGURE 1 0 x 1 _1 1 ƒ= and some partial sums 1-x v EXAMPLE 1 Express 1͑͞1 ϩ x 2 ͒ as the sum of a power series and ﬁnd the interval of convergence. SOLUTION Replacing x by Ϫx 2 in Equation 1, we have 1 1 ϱ ෇ ෇ ͑Ϫ 2 ͒n 2 2 ͚ x 1 ϩ x 1 Ϫ ͑Ϫx ͒ n෇0 ϱ ෇ ͚ ͑Ϫ1͒nx 2n ෇ 1 Ϫ x 2 ϩ x 4 Ϫ x 6 ϩ x 8 Ϫиии n෇0

Because this is a geometric series, it converges when Խ Ϫx 2 Խ Ͻ 1 , that is, x 2 Ͻ 1 , or Խ x Խ Ͻ 1. Therefore the interval of convergence is ͑Ϫ1, 1͒ . (Of course, we could have determined the radius of convergence by applying the Ratio Test, but that much work is unnecessary here.)

EXAMPLE 2 Find a power series representation for 1͑͞x ϩ 2͒ . SOLUTION In order to put this function in the form of the left side of Equation 1, we ﬁrst factor a 2 from the denominator:

1 ෇ 1 ෇ 1 2 ϩ x x x 2ͩ1 ϩ ͪ 2ͫ1 Ϫ ͩϪ ͪͬ 2 2

1 ϱ x n ϱ ͑Ϫ1͒n ෇ ͩϪ ͪ ෇ ͚ n ͚ nϩ1 x 2 n෇0 2 n෇0 2

This series converges when Խ Ϫx͞2 Խ Ͻ 1, that is, Խ x Խ Ͻ 2 . So the interval of convergence is ͑Ϫ2, 2͒ .

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748 CHAPTER 11 INFINITE SEQUENCES AND SERIES

EXAMPLE 3 Find a power series representation of x 3͑͞x ϩ 2͒ . SOLUTION Since this function is just x 3 times the function in Example 2, all we have to do is to multiply that series by x 3 :

It’s legitimate to move x 3 across the x 3 1 ϱ ͑Ϫ1͒n ϱ ͑Ϫ1͒n sigma sign because it doesn’t depend on . ෇ x 3 ؒ ෇ x 3 ͚ x n ෇ ͚ x nϩ3 n ϩ ϩ nϩ1 nϩ1 [Use Theorem 11.2.8(i) with c ෇ x 3 .] x 2 x 2 n෇0 2 n෇0 2 ෇ 1 3 Ϫ 1 4 ϩ 1 5 Ϫ 1 6 ϩиии 2 x 4 x 8 x 16 x

Another way of writing this series is as follows:

x 3 ϱ ͑Ϫ1͒nϪ1 ෇ n ͚ nϪ2 x x ϩ 2 n෇3 2

As in Example 2, the interval of convergence is ͑Ϫ2, 2͒ . Differentiation and Integration of Power Series ͸ϱ n The sum of a power series is a function f ͑x͒ ෇ n෇0 cn͑x Ϫ a͒ whose domain is the interval of convergence of the series. We would like to be able to differentiate and integrate such functions, and the following theorem (which we won’t prove) says that we can do so by differentiating or integrating each individual term in the series, just as we would for a polynomial. This is called term-by-term differentiation and integration.

͸ n 2 Theorem If the power series cn͑x Ϫ a͒ has radius of convergence R Ͼ 0 , then the function f deﬁned by

ϱ 2 n f ͑x͒ ෇ c0 ϩ c1͑x Ϫ a͒ ϩ c2͑x Ϫ a͒ ϩиии෇ ͚ cn͑x Ϫ a͒ n෇0 is differentiable (and therefore continuous) on the interval ͑a Ϫ R, a ϩ R͒ and

ϱ 2 nϪ1 (i) f Ј͑x͒ ෇ c1 ϩ 2c2͑x Ϫ a͒ ϩ 3c3͑x Ϫ a͒ ϩиии෇ ͚ ncn͑x Ϫ a͒ n෇1 ͑x Ϫ a͒2 ͑x Ϫ a͒3 In part (ii), x c dx ෇ c x ϩ C is written as (ii) y f ͑x͒ dx ෇ C ϩ c ͑x Ϫ a͒ ϩ c ϩ c ϩиии 0 0 1 0 1 2 2 3 c0͑x Ϫ a͒ ϩ C, where C ෇ C1 ϩ ac0, so all the terms of the series have the same form. ϱ ͑x Ϫ a͒nϩ1 ෇ C ϩ ͚ cn n෇0 n ϩ 1 The radii of convergence of the power series in Equations (i) and (ii) are both R .

NOTE 1 Equations (i) and (ii) in Theorem 2 can be rewritten in the form

d ϱ ϱ d ͚ͫ ͑ Ϫ ͒nͬ ෇ ͚ ͓ ͑ Ϫ ͒n ͔ (iii) cn x a cn x a dx n෇0 n෇0 dx

ϱ ϱ ͚ͫ ͑ Ϫ ͒nͬ ෇ ͚ y ͑ Ϫ ͒n (iv) y cn x a dx cn x a dx n෇0 n෇0

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SECTION 11.9 REPRESENTATIONS OF FUNCTIONS AS POWER SERIES 749

We know that, for ﬁnite sums, the derivative of a sum is the sum of the derivatives and the integral of a sum is the sum of the integrals. Equations (iii) and (iv) assert that the same is true for inﬁnite sums, provided we are dealing with power series. (For other types of series of functions the situation is not as simple; see Exercise 38.)

NOTE 2 Although Theorem 2 says that the radius of convergence remains the same when a power series is differentiated or integrated, this does not mean that the interval of convergence remains the same. It may happen that the original series converges at an endpoint, whereas the differentiated series diverges there. (See Exercise 39.)

NOTE 3 The idea of differentiating a power series term by term is the basis for a power- ful method for solving differential equations. We will discuss this method in Chapter 17.

EXAMPLE 4 In Example 3 in Section 11.8 we saw that the Bessel function

ϱ ͑Ϫ ͒n 2n ͑ ͒ ෇ 1 x J0 x ͚ 2n 2 n෇0 2 ͑n!͒

is deﬁned for all x . Thus, by Theorem 2, J0 is differentiable for all x and its derivative is found by term-by-term differentiation as follows:

ϱ ͑Ϫ ͒n 2n ϱ ͑Ϫ ͒n 2nϪ1 Ј͑ ͒ ෇ d 1 x ෇ 1 2nx J0 x ͚ 2n 2 ͚ 2n 2 n෇0 dx 2 ͑n!͒ n෇1 2 ͑n!͒

v EXAMPLE 5 Express 1͑͞1 Ϫ x͒2 as a power series by differentiating Equation 1. What is the radius of convergence? SOLUTION Differentiating each side of the equation 1 ϱ ෇ 1 ϩ x ϩ x 2 ϩ x 3 ϩиии෇ ͚ x n 1 Ϫ x n෇0 1 ϱ ෇ ϩ ϩ 2 ϩиии෇ nϪ1 we get 2 1 2x 3x ͚ nx ͑1 Ϫ x͒ n෇1 If we wish, we can replace n by n ϩ 1 and write the answer as 1 ϱ ෇ ͑ ϩ ͒ n 2 ͚ n 1 x ͑1 Ϫ x͒ n෇0

According to Theorem 2, the radius of convergence of the differentiated series is the same as the radius of convergence of the original series, namely, R ෇ 1 .

EXAMPLE 6 Find a power series representation for ln͑1 ϩ x͒ and its radius of convergence. SOLUTION We notice that the derivative of this function is 1͑͞1 ϩ x͒ . From Equation 1 we have 1 1 ෇ ෇ 1 Ϫ x ϩ x 2 Ϫ x 3 ϩиии Խ x Խ Ͻ 1 1 ϩ x 1 Ϫ ͑Ϫx͒

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750 CHAPTER 11 INFINITE SEQUENCES AND SERIES

Integrating both sides of this equation, we get 1 ln͑1 ϩ x͒ ෇ y dx ෇ y ͑1 Ϫ x ϩ x 2 Ϫ x 3 ϩиии͒ dx 1 ϩ x

x 2 x 3 x 4 ෇ x Ϫ ϩ Ϫ ϩиииϩC 2 3 4 ϱ x n ෇ ͚ ͑Ϫ1͒nϪ1 ϩ C Խ x Խ Ͻ 1 n෇1 n

To determine the value of C we put x ෇ 0 in this equation and obtain ln͑1 ϩ 0͒ ෇ C. Thus C ෇ 0 and

x 2 x 3 x 4 ϱ x n ln͑1 ϩ x͒ ෇ x Ϫ ϩ Ϫ ϩиии෇ ͚ ͑Ϫ1͒nϪ1 Խ x Խ Ͻ 1 2 3 4 n෇1 n

The radius of convergence is the same as for the original series: R ෇ 1 .

v EXAMPLE 7 Find a power series representation for f ͑x͒ ෇ tanϪ1x. SOLUTION We observe that f Ј͑x͒ ෇ 1͑͞1 ϩ x 2 ͒ and ﬁnd the required series by integrating the power series for 1͑͞1 ϩ x 2 ͒ found in Example 1.

1 tanϪ1x ෇ y dx ෇ y ͑1 Ϫ x 2 ϩ x 4 Ϫ x 6 ϩиии͒ dx 1 ϩ x 2 The power series for tanϪ1x obtained in Example 7 is called Gregory’s series after x 3 x 5 x 7 the Scottish mathematician James Gregory ෇ C ϩ x Ϫ ϩ Ϫ ϩиии (1638–1675), who had anticipated some of 3 5 7 Newton’s discoveries. We have shown that Gregory’s series is valid when Ϫ1 Ͻ x Ͻ 1, To ﬁnd CC we put x ෇ 0 and obtain ෇ tanϪ1 0 ෇ 0. Therefore but it turns out (although it isn’t easy to prove) that it is also valid when x ෇ Ϯ1 . Notice that x 3 x 5 x 7 ϱ x 2nϩ1 when x ෇ 1 the series becomes tanϪ1x ෇ x Ϫ ϩ Ϫ ϩиии෇ ͚ ͑Ϫ1͒n 3 5 7 ෇ 2n ϩ 1 ␲ 1 1 1 n 0 ෇ 1 Ϫ ϩ Ϫ ϩиии 4 3 5 7 This beautiful result is known as the Leibniz Since the radius of convergence of the series for 1͑͞1 ϩ x 2 ͒ is 1, the radius of conver- formula for ␲ . gence of this series for tanϪ1x is also 1.

EXAMPLE 8 (a) Evaluate x ͓1͑͞1 ϩ x 7 ͔͒ dx as a power series. x0.5 ͓ ͑͞ ϩ 7 ͔͒ Ϫ7 (b) Use part (a) to approximate 0 1 1 x dx correct to within 10 . SOLUTION (a) The ﬁrst step is to express the integrand, 1͑͞1 ϩ x 7 ͒ , as the sum of a power series. As in Example 1, we start with Equation 1 and replace x by Ϫx 7 :

1 1 ϱ ෇ ෇ ͑Ϫ 7 ͒n 7 7 ͚ x 1 ϩ x 1 Ϫ ͑Ϫx ͒ n෇0

ϱ ෇ ͚ ͑Ϫ1͒nx 7n ෇ 1 Ϫ x 7 ϩ x 14 Ϫиии n෇0

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SECTION 11.9 REPRESENTATIONS OF FUNCTIONS AS POWER SERIES 751

This example demonstrates one way in Now we integrate term by term: which power series representations are useful. Integrating 1͑͞1 ϩ x 7 ͒ by hand is incredibly 1 ϱ ϱ x 7nϩ1 difficult. Different computer algebra systems y dx ෇ y ͚ ͑Ϫ1͒nx 7n dx ෇ C ϩ ͚ ͑Ϫ1͒n ϩ 7 ϩ return different forms of the answer, but they 1 x n෇0 n෇0 7n 1 are all extremely complicated. (If you have a 8 15 22 CAS, try it yourself.) The infinite series answer x x x ෇ C ϩ x Ϫ ϩ Ϫ ϩиии that we obtain in Example 8(a) is actually much 8 15 22 easier to deal with than the finite answer provided by a CAS. This series converges for Խ Ϫx 7 Խ Ͻ 1 , that is, for Խ x Խ Ͻ 1 . (b) In applying the Fundamental Theorem of Calculus, it doesn’t matter which antiderivative we use, so let’s use the antiderivative from part (a) with C ෇ 0 :

͞ 8 15 22 1 2 0.5 1 ෇ ͫ Ϫ x ϩ x Ϫ x ϩиииͬ y 7 dx x 0 ϩ 1 x 8 15 22 0 ͑Ϫ ͒n ෇ 1 Ϫ 1 ϩ 1 Ϫ 1 ϩиииϩ 1 ϩиии 2 8 и 28 15 и 215 22 и 222 ͑7n ϩ 1͒27nϩ1

This inﬁnite series is the exact value of the deﬁnite integral, but since it is an alternating series, we can approximate the sum using the Alternating Series Estimation Theorem. If we stop adding after the term with n ෇ 3 , the error is smaller than the term with n ෇ 4 : 1 Ϸ 6.4 ϫ 10Ϫ11 29 и 229 So we have

0.5 1 Ϸ 1 Ϫ 1 ϩ 1 Ϫ 1 Ϸ y 7 dx 8 15 22 0.49951374 ϩ x 2 8 ؒ 2 15 ؒ 2 22 ؒ 2 1 0

11.9 Exercises

1. ͸ϱ n x x If the radius of convergence of the power series n෇0 cn x 7. f ͑x͒ ෇ 8. f ͑x͒ ෇ is 10, what is the radius of convergence of the series 9 ϩ x 2 2x 2 ϩ 1 ͸ϱ nϪ1 n෇1 ncn x ? Why? 1 ϩ x x 2 9. ͑ ͒ ෇ 10. ͑ ͒ ෇ f x f x 3 3 ͸ϱ n 1 Ϫ x a Ϫ x 2. Suppose you know that the series n෇0 bn x converges for Խ x Խ Ͻ 2. What can you say about the following series? Why? 11–12 Express the function as the sum of a power series by ﬁrst ϱ bn ϩ using partial fractions. Find the interval of convergence. ͚ x n 1 ෇ n ϩ 1 n 0 3 x ϩ 2 11. f ͑x͒ ෇ 12. f ͑x͒ ෇ x 2 Ϫ x Ϫ 2 2x 2 Ϫ x Ϫ 1 3–10 Find a power series representation for the function and determine the interval of convergence. 1 5 13. (a) Use differentiation to ﬁnd a power series representation for 3. f ͑x͒ ෇ 4. f ͑x͒ ෇ 1 ϩ x 1 Ϫ 4x 2 1 f ͑x͒ ෇ ͑1 ϩ x͒2 2 1 5. f ͑x͒ ෇ 6. f ͑x͒ ෇ 3 Ϫ x x ϩ 10 What is the radius of convergence?

; Graphing calculator or computer required 1. Homework Hints available at stewartcalculus.com

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752 CHAPTER 11 INFINITE SEQUENCES AND SERIES

(b) Use part (a) to find a power series for 33. Use the result of Example 7 to compute arctan 0.2 correct to five decimal places. 1 f ͑x͒ ෇ ͑1 ϩ x͒3 34. Show that the function ϱ ͑Ϫ1͒nx 2n (c) Use part (b) to find a power series for f ͑x͒ ෇ ͚ n෇0 ͑2n͒! x 2 f ͑x͒ ෇ is a solution of the differential equation ͑1 ϩ x͒3 f Љ͑x͒ ϩ f ͑x͒ ෇ 0 14. (a) Use Equation 1 to find a power series representation for ͑ ͒ ෇ ͑ Ϫ ͒ f x ln 1 x . What is the radius of convergence? 35. (a) Show that J0 (the Bessel function of order 0 given in (b) Use part (a) to find a power series for f ͑x͒ ෇ x ln͑1 Ϫ x͒. Example 4) satisfies the differential equation ෇ 1 (c) By putting x 2 in your result from part (a), express ln 2 2 2 as the sum of an infinite series. x J0Љ͑x͒ ϩ xJ0Ј͑x͒ ϩ x J0͑x͒ ෇ 0

(b) Evaluate x1 J ͑x͒ dx correct to three decimal places. 15–20 Find a power series representation for the function and 0 0 determine the radius of convergence. 36. The Bessel function of order 1 is deﬁned by 15. f ͑x͒ ෇ ln͑5 Ϫ x͒ 16. f ͑x͒ ෇ x 2 tanϪ1͑x 3͒ ϱ ͑Ϫ1͒n x 2nϩ1 J ͑x͒ ෇ ͚ 3 1 ͑ ϩ ͒ 2nϩ1 x x n෇0 n! n 1 !2 17. f ͑x͒ ෇ 18. f ͑x͒ ෇ ͩ ͪ ͑1 ϩ 4x͒2 2 Ϫ x (a) Show that J1 satisﬁes the differential equation

2 1 ϩ x x ϩ x 2 Љ͑ ͒ ϩ Ј͑ ͒ ϩ ͑ 2 Ϫ ͒ ͑ ͒ ෇ 19. f ͑x͒ ෇ 20. f ͑x͒ ෇ x J1 x xJ1 x x 1 J1 x 0 ͑1 Ϫ x͒2 ͑1 Ϫ x͒3

(b) Show that J0Ј͑x͒ ෇ ϪJ1͑x͒.

; 21–24 Find a power series representation forf , and graph f and 37. (a) Show that the function s ͑ ͒ n ϱ several partial sums n x on the same screen. What happens as x n increases? f ͑x͒ ෇ ͚ n෇0 n! x 21. f ͑x͒ ෇ 22. f ͑x͒ ෇ ln͑x 2 ϩ 4͒ x 2 ϩ 16 is a solution of the differential equation f Ј͑x͒ ෇ f ͑x͒ 1 ϩ x 23. f ͑x͒ ෇ lnͩ ͪ 24. f ͑x͒ ෇ tanϪ1͑2x͒ 1 Ϫ x (b) Show that f ͑x͒ ෇ e x .

2 ͸ 38. Let fn͑x͒ ෇ ͑sin nx͒͞n . Show that the series fn͑x͒ 25–28 Evaluate the indeﬁnite integral as a power series. What is converges for all values of x but the series of derivatives ͸ the radius of convergence? fnЈ͑x͒ diverges when x ෇ 2n␲ , n an integer. For what values of x does the series ͸ f Љ͑x͒ converge? t t n 25. y dt 26. y dt 1 Ϫ t 8 1 ϩ t 3 39. Let ϱ n Ϫ x tan 1x f ͑x͒ ෇ ͚ 27. y x 2 ln͑1 ϩ x͒ dx 28. y dx 2 x n෇1 n Find the intervals of convergence for ff , Ј , and f Љ .

40. ͸ϱ n 29–32 Use a power series to approximate the deﬁnite integral to (a) Starting with the geometric series n෇0 x , ﬁnd the sum of six decimal places. the series ϱ nϪ1 0.2 1 0.4 Ͻ 29. 30. ͑ ϩ 4͒ ͚ nx Խ x Խ 1 y 5 dx y ln 1 x dx n෇1 0 1 ϩ x 0

2 (b) Find the sum of each of the following series. 0.1 0.3 x 31. ͑ ͒ 32. ϱ ϱ y x arctan 3x dx y 4 dx n 0 0 ϩ n Ͻ 1 x (i)͚ nx , Խ x Խ 1 (ii) ͚ n n෇1 n෇1 2

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SECTION 11.10 TAYLOR AND MACLAURIN SERIES 753

(c) Find the sum of each of the following series. 42. (a) By completing the square, show that ϱ n ␲ (i) ͚ n͑n Ϫ 1͒x , Խ x Խ Ͻ 1 1͞2 dx ෇ ෇ y 2 n 2 0 x Ϫ x ϩ 1 3s3 ϱ n2 Ϫ n ϱ n2 3 ϩ (ii) ͚ n (iii) ͚ n (b) By factoring x 1 as a sum of cubes, rewrite the inte- ෇ 2 ෇ 2 n 2 n 1 gral in part (a). Then express 1͑͞x 3 ϩ 1͒ as the sum of a 41. Use the power series for tan Ϫ1x to prove the following power series and use it to prove the following formula expression for ␲ as the sum of an inﬁnite series: for ␲:

ϱ ͑Ϫ1͒n 3s3 ϱ ͑Ϫ1͒n 2 1 ␲ ෇ s ␲ ෇ ͩ ϩ ͪ 2 3 ͚ n ͚ n n෇0 ͑2n ϩ 1͒3 4 n෇0 8 3n ϩ 1 3n ϩ 2

11.10 Taylor and Maclaurin Series

In the preceding section we were able to ﬁnd power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions have power series representations? How can we ﬁnd such representations? We start by supposing that f is any function that can be represented by a power series

2 3 4 1 f ͑x͒ ෇ c0 ϩ c1͑x Ϫ a͒ ϩ c2͑x Ϫ a͒ ϩ c3͑x Ϫ a͒ ϩ c4͑x Ϫ a͒ ϩиии Խ x Ϫ a Խ Ͻ R

Let’s try to determine what the coefﬁcients cn must be in terms of f . To begin, notice that if we put x ෇ a in Equation 1, then all terms after the ﬁrst one are 0 and we get

f ͑a͒ ෇ c0

By Theorem 11.9.2, we can differentiate the series in Equation 1 term by term:

2 3 2 f Ј͑x͒ ෇ c1 ϩ 2c2͑x Ϫ a͒ ϩ 3c3͑x Ϫ a͒ ϩ 4c4͑x Ϫ a͒ ϩиии Խ x Ϫ a Խ Ͻ R

and substitution of x ෇ a in Equation 2 gives

f Ј͑a͒ ෇ c1

Now we differentiate both sides of Equation 2 and obtain

2 f Љ͑x͒ ෇ 2c2 ϩ 2 ؒ 3c3͑x Ϫ a͒ ϩ 3 ؒ 4c4͑x Ϫ a͒ ϩиии Խ x Ϫ a Խ Ͻ R 3

Again we put x ෇ a in Equation 3. The result is

f Љ͑a͒ ෇ 2c2

Let’s apply the procedure one more time. Differentiation of the series in Equation 3 gives

2 f ٞ͑x͒ ෇ 2 ؒ 3c3 ϩ 2 ؒ 3 ؒ 4c4͑x Ϫ a͒ ϩ 3 ؒ 4 ؒ 5c5͑x Ϫ a͒ ϩиии Խ x Ϫ a Խ Ͻ R 4

and substitution of x ෇ a in Equation 4 gives f ٞ͑a͒ ෇ 2 ؒ 3c3 ෇ 3!c3

By now you can see the pattern. If we continue to differentiate and substitute x ෇ a , we obtain ͑n͒ f ͑a͒ ෇ 2 ؒ 3 ؒ 4 ؒ иии ؒ ncn ෇ n!cn

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754 CHAPTER 11 INFINITE SEQUENCES AND SERIES

Solving this equation for the nth coefﬁcient cn, we get

f ͑n͒͑a͒ c ෇ n n!

This formula remains valid even for n ෇ 0 if we adopt the conventions that 0! ෇ 1 and f ͑0͒ ෇ f. Thus we have proved the following theorem.

5 Theorem If fa has a power series representation (expansion) at , that is, if

ϱ n f ͑x͒ ෇ ͚ cn͑x Ϫ a͒ Խ x Ϫ a Խ Ͻ R n෇0

then its coefﬁcients are given by the formula

f ͑n͒͑a͒ c ෇ n n!

Substituting this formula for cn back into the series, we see that if f has a power series expansion at a , then it must be of the following form.

ϱ f ͑n͒͑a͒ 6 f ͑x͒ ෇ ͚ ͑x Ϫ a͒n n෇0 n! f Ј͑a͒ f Љ͑a͒ f ٞ͑a͒ ෇ f ͑a͒ ϩ ͑x Ϫ a͒ ϩ ͑x Ϫ a͒2 ϩ ͑x Ϫ a͒3 ϩиии 1! 2! 3!

Taylor and Maclaurin The series in Equation 6 is called the Taylor series of the function f at a (or about a or The Taylor series is named after the English centered at a). For the special case a ෇ 0 the Taylor series becomes mathematician Brook Taylor (1685–1731) and the Maclaurin series is named in honor of the Scottish mathematician Colin Maclaurin ϱ f ͑n͒͑0͒ f Ј͑0͒ f Љ͑0͒ (1698–1746) despite the fact that the Maclaurin 7 f ͑x͒ ෇ ͚ x n ෇ f ͑0͒ ϩ x ϩ x 2 ϩиии series is really just a special case of the Taylor n෇0 n! 1! 2! series. But the idea of representing particular functions as sums of power series goes back to Newton, and the general Taylor series was known to the Scottish mathematician This case arises frequently enough that it is given the special name Maclaurin series. James Gregory in 1668 and to the Swiss mathematician John Bernoulli in the 1690s. NOTE We have shown that if f can be represented as a power series about , then fa is Taylor was apparently unaware of the work of equal to the sum of its Taylor series. But there exist functions that are not equal to the sum Gregory and Bernoulli when he published his of their Taylor series. An example of such a function is given in Exercise 74. discoveries on series in 1715 in his book . Methodus incrementorum directa et inversa v EXAMPLE 1 ͑ ͒ ෇ x Maclaurin series are named after Colin Maclau- Find the Maclaurin series of the function f x e and its radius of rin because he popularized them in his calculus convergence. textbook Treatise of Fluxions published in 1742. ͑ ͒ ͑ ͒ SOLUTION If f ͑x͒ ෇ e x , then f n ͑x͒ ෇ e x , so f n ͑0͒ ෇ e 0 ෇ 1 for all n . Therefore the Taylor series for f at 0 (that is, the Maclaurin series) is

ϱ f ͑n͒͑0͒ ϱ x n x x 2 x 3 ͚ x n ෇ ͚ ෇ 1 ϩ ϩ ϩ ϩиии n෇0 n! n෇0 n! 1! 2! 3!

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SECTION 11.10 TAYLOR AND MACLAURIN SERIES 755

n To ﬁnd the radius of convergence we let an ෇ x ͞n! . Then

nϩ1 anϩ1 x n! Խ x Խ Ϳ Ϳ ෇ Ϳ ؒ Ϳ ෇ l Ͻ n 0 1 an ͑n ϩ 1͒! x n ϩ 1

so, by the Ratio Test, the series converges for all x and the radius of convergence is R ෇ ϱ.

The conclusion we can draw from Theorem 5 and Example 1 is that if e x has a power series expansion at 0, then ϱ x n e x ෇ ͚ n෇0 n!

So how can we determine whether e x does have a power series representation? Let’s investigate the more general question: Under what circumstances is a function equal to the sum of its Taylor series? In other words, if f has derivatives of all orders, when is it true that ϱ f ͑n͒͑a͒ f ͑x͒ ෇ ͚ ͑x Ϫ a͒n n෇0 n!

As with any convergent series, this means that f ͑x͒ is the limit of the sequence of partial sums. In the case of the Taylor series, the partial sums are

n ͑i͒͑ ͒ f a i Tn͑x͒ ෇ ͚ ͑x Ϫ a͒ i෇0 i! f Ј͑a͒ f Љ͑a͒ f ͑n͒͑a͒ ෇ f ͑a͒ ϩ ͑x Ϫ a͒ ϩ ͑x Ϫ a͒2 ϩиииϩ ͑x Ϫ a͒n 1! 2! n!

y Notice that Tn is a polynomial of degree n called the nth-degree Taylor polynomial of f at ͑ ͒ ෇ x y=´ a. For instance, for the exponential function f x e , the result of Example 1 shows that the Taylor polynomials at 0 (or Maclaurin polynomials) with n ෇ 1 , 2, and 3 are y=T£(x) y=T™(x) x 2 x 2 x 3 T ͑x͒ ෇ 1 ϩ xT͑x͒ ෇ 1 ϩ x ϩ T ͑x͒ ෇ 1 ϩ x ϩ ϩ y=T™(x) 1 2 2! 3 2! 3! (0, 1) y=T¡(x) The graphs of the exponential function and these three Taylor polynomials are drawn in 0 x Figure 1. In general, f ͑x͒ is the sum of its Taylor series if y=T£(x) f ͑x͒ ෇ lim Tn͑x͒ FIGURE 1 n l ϱ If we let x As n increases, Tn͑x͒ appears to approach e in x Figure 1. This suggests that e is equal to the Rn͑x͒ ෇ f ͑x͒ Ϫ Tn͑x͒ so that f ͑x͒ ෇ Tn͑x͒ ϩ Rn͑x͒ sum of its Taylor series.

then Rn͑x͒ is called the remainder of the Taylor series. If we can somehow show that lim n l ϱ Rn͑x͒ ෇ 0, then it follows that

lim Tn͑x͒ ෇ lim ͓ f ͑x͒ Ϫ Rn͑x͔͒ ෇ f ͑x͒ Ϫ lim Rn͑x͒ ෇ f ͑x͒ n l ϱ n l ϱ n l ϱ

We have therefore proved the following theorem.

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756 CHAPTER 11 INFINITE SEQUENCES AND SERIES

8 Theorem If f ͑x͒ ෇ Tn͑x͒ ϩ Rn͑x͒, where Tn is the nth-degree Taylor polynomial of at af and

lim Rn͑x͒ ෇ 0 n l ϱ

for Խ x Ϫ a Խ Ͻ R, then f is equal to the sum of its Taylor series on the interval Խ x Ϫ a Խ Ͻ R.

In trying to show that lim n l ϱ Rn͑x͒ ෇ 0 for a speciﬁc functionf , we usually use the following theorem.

9 Taylor’s Inequality If Խ f ͑nϩ1͒͑x͒ Խ ഛ M for Խ x Ϫ a Խ ഛ d, then the remainder

Rn͑x͒ of the Taylor series satisﬁes the inequality

M nϩ1 Խ Rn͑x͒ Խ ഛ Խ x Ϫ a Խ for Խ x Ϫ a Խ ഛ d ͑n ϩ 1͒!

To see why this is true for n ෇ 1, we assume that Խ f Љ͑x͒ Խ ഛ M. In particular, we have f Љ͑x͒ ഛ M, so for a ഛ x ഛ a ϩ d we have

x x y f Љ͑t͒ dt ഛ y M dt a a

Formulas for the Taylor Remainder Term An antiderivative of f Љ is f Ј , so by Part 2 of the Fundamental Theorem of Calculus, we As alternatives to Taylor’s Inequality, we have have the following formulas for the remainder term. f Ј͑x͒ Ϫ f Ј͑a͒ ഛ M͑x Ϫ a͒ or f Ј͑x͒ ഛ f Ј͑a͒ ϩ M͑x Ϫ a͒ If f ͑nϩ1͒ is continuous on an interval I and x ʦ I, then x x x Ј͑ ͒ ഛ ͓ Ј͑ ͒ ϩ ͑ Ϫ ͔͒ 1 n ͑nϩ1͒ Thus y f t dt y f a M t a dt Rn͑x͒ ෇ y ͑x Ϫ t͒ f ͑t͒ dt a a n! a This is called the ͑x Ϫ a͒2 integral form of the remainder f ͑x͒ Ϫ f ͑a͒ ഛ f Ј͑a͒͑x Ϫ a͒ ϩ M term. Another formula, called Lagrange’s form 2 of the remainder term, states that there is a number z between and ax such that M f ͑x͒ Ϫ f ͑a͒ Ϫ f Ј͑a͒͑x Ϫ a͒ ഛ ͑x Ϫ a͒2 f ͑nϩ1͒͑z͒ 2 R ͑x͒ ෇ ͑x Ϫ a͒nϩ1 n ͑n ϩ 1͒! ͑ ͒ ෇ ͑ ͒ Ϫ ͑ ͒ ෇ ͑ ͒ Ϫ ͑ ͒ Ϫ Ј͑ ͒͑ Ϫ ͒ This version is an extension of the Mean Value But R1 x f x T1 x f x f a f a x a . So Theorem (which is the case n ෇ 0 ). Proofs of these formulas, together with M R ͑x͒ ഛ ͑x Ϫ a͒2 discussions of how to use them to solve the 1 2 examples of Sections 11.10 and 11.11, are given on the website A similar argument, using f Љ͑x͒ ജϪM, shows that www.stewartcalculus.com

Click on and then on M 2 Additional Topics Formulas R1͑x͒ ജϪ ͑x Ϫ a͒ for the Remainder Term in Taylor series. 2

M 2 So Խ R1͑x͒ Խ ഛ Խ x Ϫ a Խ 2

Although we have assumed that x Ͼ a , similar calculations show that this inequality is also true for x Ͻ a .

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SECTION 11.10 TAYLOR AND MACLAURIN SERIES 757

This proves Taylor’s Inequality for the case where n ෇ 1 . The result for any n is proved in a similar way by integrating n ϩ 1 times. (See Exercise 73 for the case n ෇ 2 .)

NOTE In Section 11.11 we will explore the use of Taylor’s Inequality in approximating functions. Our immediate use of it is in conjunction with Theorem 8.

In applying Theorems 8 and 9 it is often helpful to make use of the following fact.

x n 10 lim ෇ 0 for every real number x n l ϱ n!

This is true because we know from Example 1 that the series ͸ x n͞n! converges for all x and so its nth term approaches 0.

v EXAMPLE 2 Prove that e x is equal to the sum of its Maclaurin series. ͑ ϩ ͒ SOLUTION If f ͑x͒ ෇ e x , then f n 1 ͑x͒ ෇ e x for all n. If d is any positive number and Խ x Խ ഛ d, then Խ f ͑nϩ1͒͑x͒ Խ ෇ e x ഛ e d. So Taylor’s Inequality, with a ෇ 0 and M ෇ e d , says that

d e nϩ1 Խ Rn͑x͒ Խ ഛ Խ x Խ for Խ x Խ ഛ d ͑n ϩ 1͒!

Notice that the same constant M ෇ e d works for every value of n. But, from Equa - tion 10, we have e d Խ x Խnϩ1 lim Խ x Խnϩ1 ෇ e d lim ෇ 0 n l ϱ ͑n ϩ 1͒! n l ϱ ͑n ϩ 1͒!

It follows from the Squeeze Theorem that lim n l ϱ Խ Rn͑x͒ Խ ෇ 0 and therefore x lim n l ϱ Rn͑x͒ ෇ 0 for all values of x. By Theorem 8, e is equal to the sum of its Maclaurin series, that is,

ϱ x n 11 e x ෇ ͚ for all x n෇0 n!

In particular, if we put x ෇ 1 in Equation 11, we obtain the following expression for the number e as a sum of an infinite series: In 1748 Leonhard Euler used Equation 12 to find the value of e correct to 23 digits. In 2007 Shigeru Kondo, again using the series in 12 , ϱ 1 1 1 1 computed e to more than 100 billion decimal 12 e ෇ ͚ ෇ 1 ϩ ϩ ϩ ϩиии places. The special techniques employed to n෇0 n! 1! 2! 3! speed up the computation are explained on the website numbers.computation.free.fr EXAMPLE 3 Find the Taylor series for f ͑x͒ ෇ e x at a ෇ 2 . ͑ ͒ SOLUTION We have f n ͑2͒ ෇ e 2 and so, putting a ෇ 2 in the definition of a Taylor series 6 , we get ϱ f ͑n͒͑2͒ ϱ e 2 ͚ ͑x Ϫ 2͒n ෇ ͚ ͑x Ϫ 2͒n n෇0 n! n෇0 n!

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758 CHAPTER 11 INFINITE SEQUENCES AND SERIES

Again it can be veriﬁed, as in Example 1, that the radius of convergence is R ෇ ϱ . As in

Example 2 we can verify that lim n l ϱ Rn͑x͒ ෇ 0, so

ϱ e 2 13 e x ෇ ͚ ͑x Ϫ 2͒n for all x n෇0 n!

We have two power series expansions for e x , the Maclaurin series in Equation 11 and the Taylor series in Equation 13. The ﬁrst is better if we are interested in values of x near 0 and the second is better if x is near 2.

EXAMPLE 4 Find the Maclaurin series for sin x and prove that it represents sin xx for all . SOLUTION We arrange our computation in two columns as follows:

f ͑x͒ ෇ sin xf͑0͒ ෇ 0

f Ј͑x͒ ෇ cos xfЈ͑0͒ ෇ 1 f Љ͑x͒ ෇ Ϫsin xfЉ͑0͒ ෇ 0 Figure 2 shows the graph of sin x together with its Taylor (or Maclaurin) polynomials f ٞ͑x͒ ෇ Ϫcos xfٞ͑0͒ ෇ Ϫ1 ͑ ͒ ෇ T1 x x f ͑4͒͑x͒ ෇ sin xf͑4͒͑0͒ ෇ 0 x 3 T3͑x͒ ෇ x Ϫ 3! Since the derivatives repeat in a cycle of four, we can write the Maclaurin series as x 3 x 5 T5͑x͒ ෇ x Ϫ ϩ follows: f Ј͑0͒ f Љ͑0͒ f ٞ͑0͒ !5 !3 ͑ ͒ ϩ ϩ 2 ϩ 3 ϩиии Notice that, as n increases, Tn͑x͒ becomes a f 0 x x x better approximation to sin x . 1! 2! 3! x 3 x 5 x 7 ϱ x 2nϩ1 y ෇ Ϫ ϩ Ϫ ϩиии෇ ͚ ͑Ϫ ͒n x 1 ͑ ϩ ͒ T¡ 3! 5! 7! n෇0 2n 1 ! 1 ͑nϩ1͒ ͑nϩ1͒ T∞ Since f ͑x͒ is Ϯsin x or Ϯcos x , we know that Խ f ͑x͒ Խ ഛ 1 for all x. So we can take M ෇ 1 in Taylor’s Inequality: y=sin x

nϩ1 0 1 x M nϩ1 Խ x Խ 14 Խ Rn͑x͒ Խ ഛ Խ x Խ ෇ ͑n ϩ 1͒! ͑n ϩ 1͒! T£ By Equation 10 the right side of this inequality approaches 0 as n l ϱ , so Խ Rn͑x͒ Խl 0 by the Squeeze Theorem. It follows that Rn͑x͒ l 0 as n l ϱ , sosin x is equal to the sum FIGURE 2 of its Maclaurin series by Theorem 8.

We state the result of Example 4 for future reference.

x 3 x 5 x 7 15 sin x ෇ x Ϫ ϩ Ϫ ϩиии 3! 5! 7! ϱ x 2nϩ1 ෇ ͚ ͑Ϫ1͒n for all x n෇0 ͑2n ϩ 1͒!

EXAMPLE 5 Find the Maclaurin series for cos x .

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SECTION 11.10 TAYLOR AND MACLAURIN SERIES 759

SOLUTION We could proceed directly as in Example 4, but it’s easier to differentiate the Maclaurin series for sin x given by Equation 15:

d d x 3 x 5 x 7 cos x ෇ ͑sin x͒ ෇ ͩx Ϫ ϩ Ϫ ϩиииͪ dx dx 3! 5! 7!

3x 2 5x 4 7x 6 x 2 x 4 x 6 ෇ 1 Ϫ ϩ Ϫ ϩиии෇ 1 Ϫ ϩ Ϫ ϩиии 3! 5! 7! 2! 4! 6!

The Maclaurin series for x , sin xe , and cos x Since the Maclaurin series for sin xx converges for all , Theorem 2 in Section 11.9 tells us that we found in Examples 2, 4, and 5 were dis- that the differentiated series for cos xx also converges for all . Thus covered, using different methods, by Newton. These equations are remarkable because they say we know everything about each of these functions if we know all its derivatives at the x 2 x 4 x 6 16 cos x ෇ 1 Ϫ ϩ Ϫ ϩиии single number 0. 2! 4! 6! ϱ x 2n ෇ ͚ ͑Ϫ1͒n for all x n෇0 ͑2n͒!

EXAMPLE 6 Find the Maclaurin series for the function f ͑x͒ ෇ x cos x. SOLUTION Instead of computing derivatives and substituting in Equation 7, it’s easier to multiply the series for cos xx (Equation 16) by :

ϱ x 2n ϱ x 2nϩ1 x cos x ෇ x ͚ ͑Ϫ1͒n ෇ ͚ ͑Ϫ1͒n n෇0 ͑2n͒! n෇0 ͑2n͒!

EXAMPLE 7 Represent f ͑x͒ ෇ sin x as the sum of its Taylor series centered at ␲͞3 . SOLUTION Arranging our work in columns, we have

␲ s3 We have obtained two different series f ͑x͒ ෇ sin xfͩ ͪ ෇ representations for sin x , the Maclaurin series 3 2 in Example 4 and the Taylor series in Example 7. It is best to use the Maclaurin series for val- ␲ 1 ues of near 0 and the Taylor series for near Ј͑ ͒ ෇ Јͩ ͪ ෇ x x f x cos xf 3 2 ␲͞3. Notice that the third Taylor polynomial T3 in Figure 3 is a good approximation to sin x ␲ s near ␲͞3 but not as good near 0. Compare it Љ͑ ͒ ෇ Ϫ Љͩ ͪ ෇ Ϫ 3 with the third Maclaurin polynomial in Fig- f x sin xf T3 3 2 ure2, where the opposite is true. ␲ 1 y f ٞ͑x͒ ෇ Ϫcos xfٞͩ ͪ ෇ Ϫ 3 2 y=sin x and this pattern repeats indeﬁnitely. Therefore the Taylor series at ␲͞3 is

␲ ␲ ␲ ͪ f Јͩ ͪ f Љͩ ͪ f ٞͩ 0 π x ␲ 3 ␲ 3 ␲ 2 3 ␲ 3 3 f ͩ ͪ ϩ ͩx Ϫ ͪ ϩ ͩx Ϫ ͪ ϩ ͩx Ϫ ͪ ϩиии 3 1! 3 2! 3 3! 3 T£ s3 1 ␲ s3 ␲ 2 1 ␲ 3 ෇ ϩ ͩx Ϫ ͪ Ϫ ͩx Ϫ ͪ Ϫ ͩx Ϫ ͪ ϩиии FIGURE 3 2 2 ؒ 1! 3 2 ؒ 2! 3 2 ؒ 3! 3

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760 CHAPTER 11 INFINITE SEQUENCES AND SERIES

The proof that this series represents sin xx for all is very similar to that in Example 4. (Just replace xx by Ϫ ␲͞3 in 14 .) We can write the series in sigma notation if we sepa- rate the terms that contain s3 :

ϱ ͑Ϫ1͒ns3 ␲ 2n ϱ ͑Ϫ1͒n ␲ 2nϩ1 sin x ෇ ͚ ͩx Ϫ ͪ ϩ ͚ ͩx Ϫ ͪ n෇0 2͑2n͒! 3 n෇0 2͑2n ϩ 1͒! 3

The power series that we obtained by indirect methods in Examples 5 and 6 and in Section 11.9 are indeed the Taylor or Maclaurin series of the given functions because ͸ n Theorem 5 asserts that, no matter how a power series representation f ͑x͒ ෇ cn͑x Ϫ a͒ is ͑n͒ obtained, it is always true that cn ෇ f ͑a͒͞n!. In other words, the coefﬁcients are uniquely determined.

EXAMPLE 8 Find the Maclaurin series for f ͑x͒ ෇ ͑1 ϩ x͒k, where k is any real number. SOLUTION Arranging our work in columns, we have

f ͑x͒ ෇ ͑1 ϩ x͒k f ͑0͒ ෇ 1

f Ј͑x͒ ෇ k͑1 ϩ x͒kϪ1 f Ј͑0͒ ෇ k

f Љ͑x͒ ෇ k͑k Ϫ 1͒͑1 ϩ x͒kϪ2 f Љ͑0͒ ෇ k͑k Ϫ 1͒ f ٞ͑x͒ ෇ k͑k Ϫ 1͒͑k Ϫ 2͒͑1 ϩ x͒kϪ3 f ٞ͑0͒ ෇ k͑k Ϫ 1͒͑k Ϫ 2͒ ...... f ͑n͒͑x͒ ෇ k͑k Ϫ 1͒ иии͑k Ϫ n ϩ 1͒͑1 ϩ x͒kϪn f ͑n͒͑0͒ ෇ k͑k Ϫ 1͒ иии͑k Ϫ n ϩ 1͒

Therefore the Maclaurin series of f ͑x͒ ෇ ͑1 ϩ x͒k is

ϱ f ͑n͒͑0͒ ϱ k͑k Ϫ 1͒ иии͑k Ϫ n ϩ 1͒ ͚ x n ෇ ͚ x n n෇0 n! n෇0 n! This series is called the binomial series. Notice that if k is a nonnegative integer, then the terms are eventually 0 and so the series is ﬁnite. For other values of k none of the

terms is 0 and so we can try the Ratio Test. If the na th term is n, then

͑ Ϫ ͒ иии͑ Ϫ ϩ ͒͑ Ϫ ͒ nϩ1 Ϳ anϩ1 Ϳ ෇ Ϳ k k 1 k n 1 k n x ؒ n! Ϳ n an ͑n ϩ 1͒! k͑k Ϫ 1͒ иии͑k Ϫ n ϩ 1͒x

k Ϳ1 Ϫ Ϳ Խ k Ϫ n Խ n ෇ Խ x Խ ෇ Խ x Խ l Խ x Խ as n l ϱ n ϩ 1 1 1 ϩ n Thus, by the Ratio Test, the binomial series converges if Խ x Խ Ͻ 1 and diverges if Խ x Խ Ͼ 1.

The traditional notation for the coefﬁcients in the binomial series is ͑ Ϫ ͒͑ Ϫ ͒ иии͑ Ϫ ϩ ͒ ͩ k ͪ ෇ k k 1 k 2 k n 1 n n! and these numbers are called the binomial coefﬁcients.

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SECTION 11.10 TAYLOR AND MACLAURIN SERIES 761

The following theorem states that ͑1 ϩ x͒k is equal to the sum of its Maclaurin series. It

is possible to prove this by showing that the remainder term Rn͑x͒ approaches 0, but that turns out to be quite difﬁcult. The proof outlined in Exercise 75 is much easier.

17 The Binomial Series If k is any real number and Խ x Խ Ͻ 1 , then

ϱ k k͑k Ϫ 1͒ k͑k Ϫ 1͒͑k Ϫ 2͒ ͑1 ϩ x͒k ෇ ͚ ͩ ͪx n ෇ 1 ϩ kx ϩ x 2 ϩ x 3 ϩ иии n෇0 n 2! 3!

Although the binomial series always converges when Խ x Խ Ͻ 1 , the question of whether or not it converges at the endpoints, Ϯ1 , depends on the value of k . It turns out that the series converges at 1 if Ϫ1 Ͻ k ഛ 0 and at both endpoints if k ജ 0 . Notice that if k is a pos- k k itive integer and n Ͼ k , then the expression for (n ) contains a factor ͑k Ϫ k͒ , so (n ) ෇ 0 for n Ͼ k. This means that the series terminates and reduces to the ordinary Binomial Theorem when k is a positive integer. (See Reference Page 1.)

1 v EXAMPLE 9 Find the Maclaurin series for the function f ͑x͒ ෇ and its radius s Ϫ of convergence. 4 x

SOLUTION We rewrite f ͑x͒ in a form where we can use the binomial series:

1 1 1 1 x Ϫ1͞2 ෇ ෇ ෇ ͩ1 Ϫ ͪ s4 Ϫ x x x 2 4 ͱ4ͩ1 Ϫ ͪ 2ͱ1 Ϫ 4 4

෇ Ϫ 1 Ϫ ͞ Using the binomial series with k 2 and with x replaced by x 4 , we have

1 1 x Ϫ1͞2 1 ϱ Ϫ 1 x n ෇ ͩ1 Ϫ ͪ ෇ ͚ ͩ 2 ͪͩϪ ͪ s4 Ϫ x 2 4 2 n෇0 n 4

1 1 x (Ϫ 1 )(Ϫ 3 ) x 2 (Ϫ 1)(Ϫ 3)(Ϫ 5) x 3 ෇ ͫ1 ϩ ͩϪ ͪͩϪ ͪ ϩ 2 2 ͩϪ ͪ ϩ 2 2 2 ͩϪ ͪ 2 2 4 2! 4 3! 4

(Ϫ 1)(Ϫ 3)(Ϫ 5) иии(Ϫ 1 Ϫ n ϩ 1) x n ϩиииϩ 2 2 2 2 ͩϪ ͪ ϩиииͬ n! 4

иии ؒ ͑2n Ϫ 1͒ ؒ 5 ؒ 3 ؒ 1 5 ؒ 3 ؒ 1 3 ؒ 1 1 1 ෇ ͫ1 ϩ x ϩ x 2 ϩ x 3 ϩиииϩ x n ϩиииͬ 2 8 2!82 3!83 n!8n

We know from 17 that this series converges when Խ Ϫx͞4 Խ Ͻ 1, that is, Խ x Խ Ͻ 4 , so the radius of convergence is R ෇ 4 .

We collect in the following table, for future reference, some important Maclaurin series that we have derived in this section and the preceding one.

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762 CHAPTER 11 INFINITE SEQUENCES AND SERIES

TABLE 1 1 ϱ ෇ ͚ x n ෇ 1 ϩ x ϩ x 2 ϩ x 3 ϩиии R ෇ 1 Important Maclaurin Series and 1 Ϫ x n෇0 Their Radii of Convergence ϱ x n x x 2 x 3 e x ෇ ͚ ෇ 1 ϩ ϩ ϩ ϩиии R ෇ ϱ n෇0 n! 1! 2! 3! ϱ x 2nϩ1 x 3 x 5 x 7 sin x ෇ ͚ ͑Ϫ1͒n ෇ x Ϫ ϩ Ϫ ϩиии R ෇ ϱ n෇0 ͑2n ϩ 1͒! 3! 5! 7! ϱ x 2n x 2 x 4 x 6 cos x ෇ ͚ ͑Ϫ1͒n ෇ 1 Ϫ ϩ Ϫ ϩиии R ෇ ϱ n෇0 ͑2n͒! 2! 4! 6! ϱ x 2nϩ1 x 3 x 5 x 7 tanϪ1x ෇ ͚ ͑Ϫ1͒n ෇ x Ϫ ϩ Ϫ ϩиии R ෇ 1 n෇0 2n ϩ 1 3 5 7 ϱ x n x 2 x 3 x 4 ln͑1 ϩ x͒ ෇ ͚ ͑Ϫ1͒nϪ1 ෇ x Ϫ ϩ Ϫ ϩиии R ෇ 1 n෇1 n 2 3 4

ϱ k k͑k Ϫ 1͒ k͑k Ϫ 1͒͑k Ϫ 2͒ ͑1 ϩ x͒k ෇ ͚ ͩ ͪx n ෇ 1 ϩ kx ϩ x 2 ϩ x 3 ϩ иии R ෇ 1 n෇0 n 2! 3!

1 1 1 1 EXAMPLE 10 Find the sum of the series Ϫ ϩ Ϫ ϩ иии. 24 ؒ 4 23 ؒ 3 22 ؒ 2 2 ؒ 1 SOLUTION With sigma notation we can write the given series as

n ϱ 1 ϱ 1 ͑Ϫ ͒nϪ1 ෇ ͑Ϫ ͒nϪ1 (2) ͚ 1 n ͚ 1 n෇1 n ؒ 2 n෇1 n ͑ ϩ ͒ ෇ 1 Then from Table 1 we see that this series matches the entry for ln 1 x with x 2 . So ϱ 1 ͑Ϫ ͒nϪ1 ෇ ϩ 1 ෇ 3 ͚ 1 n ln(1 2) ln 2 n෇1 n ؒ 2

TEC Module 11.10/11.11 enables you to see One reason that Taylor series are important is that they enable us to integrate functions how successive Taylor polynomials approach the that we couldn’t previously handle. In fact, in the introduction to this chapter we mentioned original function. that Newton often integrated functions by ﬁrst expressing them as power series and then 2 integrating the series term by term. The function f ͑x͒ ෇ eϪx can’t be integrated by techniques discussed so far because its antiderivative is not an elementary function (see Sec- tion 7.5). In the following example we use Newton’s idea to integrate this function.

v EXAMPLE 11

(a) Evaluate x eϪx2 dx as an infinite series. x1 Ϫx2 (b) Evaluate 0 e dx correct to within an error of 0.001 . SOLUTION (a) First we find the Maclaurin series for f ͑x͒ ෇ eϪx2 . Although it’s possible to use the direct method, let’s find it simply by replacing x with Ϫx 2 in the series for e x given in Table 1. Thus, for all values of x,

ϱ 2 n ϱ 2n 2 4 6 2 ͑Ϫx ͒ x x x x eϪx ෇ ͚ ෇ ͚ ͑Ϫ1͒n ෇ 1 Ϫ ϩ Ϫ ϩиии n෇0 n! n෇0 n! 1! 2! 3!

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SECTION 11.10 TAYLOR AND MACLAURIN SERIES 763

Now we integrate term by term:

2 4 6 2n 2 x x x x y eϪx dx ෇ y ͩ1 Ϫ ϩ Ϫ ϩиииϩ͑Ϫ1͒n ϩиииͪ dx 1! 2! 3! n!

x 3 x 5 x 7 x 2nϩ1 ෇ C ϩ x Ϫ ϩ Ϫ ϩиииϩ͑Ϫ1͒n ϩиии !2n ϩ 1͒n͑ !3 ؒ 7 !2 ؒ 5 !1 ؒ 3

This series converges for all xe because the original series for Ϫx2 converges for all x . (b) The Fundamental Theorem of Calculus gives

3 5 7 9 1 1 2 x x x x y eϪx dx ෇ ͫx Ϫ ϩ Ϫ ϩ Ϫиииͬ ؒ ؒ ؒ ؒ 0 3 1! 5 2! 7 3! 9 4! 0 We can take ෇ in the antiderivative C 0 ෇ Ϫ 1 ϩ 1 Ϫ 1 ϩ 1 Ϫиии in part (a). 1 3 10 42 216 Ϸ Ϫ 1 ϩ 1 Ϫ 1 ϩ 1 Ϸ 1 3 10 42 216 0.7475

The Alternating Series Estimation Theorem shows that the error involved in this approximation is less than 1 1 ෇ Ͻ 0.001 1320 !5 ؒ 11

Another use of Taylor series is illustrated in the next example. The limit could be found with l’Hospital’s Rule, but instead we use a series.

e x Ϫ 1 Ϫ x EXAMPLE 12 Evaluate lim . x l 0 x 2

SOLUTION Using the Maclaurin series for e x , we have

x x 2 x 3 ͩ1 ϩ ϩ ϩ ϩиииͪ Ϫ 1 Ϫ x e x Ϫ 1 Ϫ x 1! 2! 3! lim ෇ lim x l 0 x 2 x l 0 x 2

2 3 4 x ϩ x ϩ x ϩ иии Some computer algebra systems compute 2! 3! 4! limits in this way. ෇ lim x l 0 x 2

1 x x 2 x 3 1 ෇ lim ͩ ϩ ϩ ϩ ϩиииͪ ෇ x l 0 2 3! 4! 5! 2

because power series are continuous functions.

Multiplication and Division of Power Series If power series are added or subtracted, they behave like polynomials (Theorem 11.2.8 shows this). In fact, as the following example illustrates, they can also be multiplied and divided like polynomials. We ﬁnd only the ﬁrst few terms because the calculations for the later terms become tedious and the initial terms are the most important ones.

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764 CHAPTER 11 INFINITE SEQUENCES AND SERIES

EXAMPLE 13 Find the ﬁrst three nonzero terms in the Maclaurin series for (a) e x sin x and (b) tan x .

SOLUTION (a) Using the Maclaurin series for e x and sin x in Table 1, we have

x x 2 x 3 x 3 e x sin x ෇ ͩ1 ϩ ϩ ϩ ϩиииͪͩx Ϫ ϩиииͪ 1! 2! 3! 3!

We multiply these expressions, collecting like terms just as for polynomials:

ϩ ϩ 1 2 ϩ 1 3 ϩиии 1 x 2 x 6 x ϫ Ϫ 1 3 ϩиии x 6 x ϩ 2 ϩ 1 3 ϩ 1 4 ϩиии x x 2 x 6 x ϩ Ϫ 1 3 Ϫ 1 4 Ϫиии 6 x 6 x ϩ 2 ϩ 1 3 ϩиии x x 3 x

x ෇ ϩ 2 ϩ 1 3 ϩиии Thus e sin x x x 3 x

(b) Using the Maclaurin series in Table 1, we have

x 3 x 5 x Ϫ ϩ Ϫиии sin x 3! 5! tan x ෇ ෇ cos x x 2 x 4 1 Ϫ ϩ Ϫиии 2! 4!

We use a procedure like long division:

ϩ 1 3 ϩ 2 5 ϩиии x 3 x 15 x Ϫ 1 2 ϩ 1 4 Ϫиии Ϫ 1 3 ϩ 1 5 Ϫиии 1 2 x 24 x )x 6 x 120 x

Ϫ 1 3 ϩ 1 5 Ϫ иии x 2 x 24 x

1 3 Ϫ 1 5 ϩиии 3 x 30 x 1 3 Ϫ 1 5 ϩиии 3 x 6 x

2 5 ϩиии 15 x

෇ ϩ 1 3 ϩ 2 5 ϩиии Thus tan x x 3 x 15 x

Although we have not attempted to justify the formal manipulations used in Exam ple 13, ͸ n they are legitimate. There is a theorem which states that if both f ͑x͒ ෇ cn x and ͸ n t͑x͒ ෇ bn x converge for Խ x Խ Ͻ R and the series are multiplied as if they were polynomials, then the resulting series also converges for Խ x Խ Ͻ Rf and represents ͑x͒t͑x͒ . For division we require b0 0 ; the resulting series converges for sufﬁciently small Խ x Խ .

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SECTION 11.10 TAYLOR AND MACLAURIN SERIES 765

11.10 Exercises

͸ϱ n 21. ␲ 1. If f ͑x͒ ෇ n෇0 bn͑x Ϫ 5͒ for all xb , write a formula for 8 . Prove that the series obtained in Exercise 7 represents sin x for all x . 2. The graph of f is shown. 22. Prove that the series obtained in Exercise 18 represents sin x y for all x . f 23. Prove that the series obtained in Exercise 11 represents sinh x for all x . 1 24. Prove that the series obtained in Exercise 12 represents cosh x for all x . 0x1 25–28 Use the binomial series to expand the function as a power series. State the radius of convergence. (a) Explain why the series 25. s4 1 Ϫ x 26. s3 8 ϩ x 2 3 1.6 Ϫ 0.8͑x Ϫ 1͒ ϩ 0.4͑x Ϫ 1͒ Ϫ 0.1͑x Ϫ 1͒ ϩиии 1 27. 28. ͑1 Ϫ x͒2͞3 ͑ ϩ ͒3 is not the Taylor series of f centered at 1. 2 x (b) Explain why the series 29–38 Use a Maclaurin series in Table 1 to obtain the Maclaurin ϩ ͑ Ϫ ͒ ϩ ͑ Ϫ ͒2 Ϫ ͑ Ϫ ͒3 ϩиии 2.8 0.5 x 2 1.5 x 2 0.1 x 2 series for the given function. is not the Taylor series of f centered at 2. 29. f ͑x͒ ෇ sin ␲xf30. ͑x͒ ෇ cos͑␲x͞2͒ ͑ ͒ 3. If f n ͑0͒ ෇ ͑n ϩ 1͒! for n ෇ 0, 1, 2, . . . , ﬁnd the Maclaurin 31. f ͑x͒ ෇ e x ϩ e 2x 32. f ͑x͒ ෇ e x ϩ 2eϪx series for f and its radius of convergence. 33. ͑ ͒ ෇ 1 2 34. ͑ ͒ ෇ 2 ͑ ϩ 3͒ f x x cos( 2 x ) f x x ln 1 x 4. Find the Taylor series for f centered at 4 if x x 2 35. f ͑x͒ ෇ 36. f ͑x͒ ෇ s ϩ 2 s ϩ ͑Ϫ1͒n n! 4 x 2 x f ͑n͒͑4͒ ෇ n͑ ϩ ͒ 37. ͑ ͒ ෇ 2 2 ෇ 1 ͑ Ϫ ͒ 3 n 1 f x sin x [Hint: Use sin x 2 1 cos 2x .]

What is the radius of convergence of the Taylor series? Ϫ x sin x 3 if x 0 5–12 ͑ ͒ 38. f ͑x͒ ෇ ͭ x Find the Maclaurin series for f x using the deﬁnition of a 1 if x ෇ 0 Maclaurin series. [Assume that f has a power series expan sion. 6

Do not show that Rn͑x͒ l 0 .] Also ﬁnd the associated radius of convergence. ; 39–42 Find the Maclaurin series of f (by any method) and its 5. f ͑x͒ ෇ ͑1 Ϫ x͒Ϫ2 6. f ͑x͒ ෇ ln͑1 ϩ x͒ radius of convergence. Graph f and its ﬁrst few Taylor polynomials Ϫ on the same screen. What do you notice about the relation ship 7. f ͑x͒ ෇ sin ␲xf8. ͑x͒ ෇ e 2x between these polynomials and f ? x 9. f ͑x͒ ෇ 2 10. f ͑x͒ ෇ x cos x 2 39. f ͑x͒ ෇ cos͑x 2 ͒ 40. f ͑x͒ ෇ eϪx ϩ cos x 11. f ͑x͒ ෇ sinh xf12. ͑x͒ ෇ cosh x 41. f ͑x͒ ෇ xeϪx 42. f ͑x͒ ෇ tanϪ1͑x 3͒

13–20 f ͑x͒ Find the Taylor series for centered at the given value 43. Use the Maclaurin series for cos x to compute cos 5Њ correct to af of . [Assume that has a power series expansion. Do not show five decimal places. that Rn͑x͒ l 0 .] Also find the associated radius of convergence. 44. Use the Maclaurin series for e x to calculate 1͞10se correct to 13. f ͑x͒ ෇ x 4 Ϫ 3x 2 ϩ 1, a ෇ 1 five decimal places. 14. f ͑x͒ ෇ x Ϫ x 3, a ෇ Ϫ2 45. (a) Use the binomial series to expand 1͞s1 Ϫ x 2 . 15. f ͑x͒ ෇ ln xa, ෇ 2 16. f ͑x͒ ෇ 1͞xa, ෇ Ϫ3 (b) Use part (a) to find the Maclaurin series for sinϪ1x . 17. f ͑x͒ ෇ e 2x, a ෇ 3 18. f ͑x͒ ෇ sin xa, ෇ ␲͞2 46. (a) Expand 1͞s4 1 ϩ x as a power series. ͞s4 19. f ͑x͒ ෇ cos xa, ෇ ␲ 20. f ͑x͒ ෇ sx , a ෇ 16 (b) Use part (a) to estimate 1 1.1 correct to three decimal places.

; Graphing calculator or computer required 1. Homework Hints available at stewartcalculus.com

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766 CHAPTER 11 INFINITE SEQUENCES AND SERIES

47–50 Evaluate the indefinite integral as an infinite series. ϱ ͑Ϫ1͒n␲ 2nϩ1 67. ͚ ϩ x 2n 1͑ ϩ ͒ e Ϫ 1 n෇0 4 2n 1 ! 47. y x cos͑x 3͒ dx 48. y dx x ͑ln 2͒2 ͑ln 2͒3 68. 1 Ϫ ln 2 ϩ Ϫ ϩиии cos x Ϫ 1 2! 3! 49. y dx 50. y arctan͑x 2͒ dx x 9 27 81 69. 3 ϩ ϩ ϩ ϩиии 2! 3! 4! 51–54 Use series to approximate the definite integral to within the 1 1 1 1 indicated accuracy. 70. Ϫ ϩ Ϫ ϩиии 27 ؒ 7 25 ؒ 5 23 ؒ 3 2 ؒ 1 1͞2 51. y x 3 arctan x dx (four decimal places) 0 71. Show that if pn is an th-degree polynomial, then 1 52. y sin͑x 4͒ dx (four decimal places) 0 n p ͑i͒͑x͒ p͑x ϩ 1͒ ෇ ͚ 0.4 i෇0 i! 53. y s1 ϩ x 4 dx (Խ error Խ Ͻ 5 ϫ 10Ϫ6) 0 72. If f ͑x͒ ෇ ͑1 ϩ x 3͒30, what is f ͑58͒͑0͒ ? 0.5 54. y x 2eϪx 2 dx (Խ error Խ Ͻ 0.001) 0 73. Prove Taylor’s Inequality for n ෇ 2 , that is, prove that if Խ f ٞ͑x͒ Խ ഛ M for Խ x Ϫ a Խ ഛ d, then

55–57 Use series to evaluate the limit. M 3 Խ R2͑x͒ Խ ഛ Խ x Ϫ a Խ for Խ x Ϫ a Խ ഛ d x Ϫ ln͑1 ϩ x͒ 1 Ϫ cos x 6 55. 56. lim lim x x l 0 x 2 x l0 1 ϩ x Ϫ e 74. (a) Show that the function deﬁned by Ϫ ϩ 1 3 sin x x 6 x 57. lim Ϫ1͞x 2 x l0 x 5 e if x 0 f ͑x͒ ෇ ͭ 0 if x ෇ 0

58. Use the series in Example 13(b) to evaluate is not equal to its Maclaurin series. ; (b) Graph the function in part (a) and comment on its behavior tan x Ϫ x lim near the origin. x l 0 x 3 75. Use the following steps to prove 17 . We found this limit in Example 4 in Section 4.4 using t͑ ͒ ෇ ͸ϱ k n (a) Let x n෇0 (n )x . Differentiate this series to show that l’Hospital’s Rule three times. Which method do you prefer? kt͑x͒ tЈ͑x͒ ෇ Ϫ1 Ͻ x Ͻ 1 59–62 Use multiplication or division of power series to ﬁnd the 1 ϩ x ﬁrst three nonzero terms in the Maclaurin series for each function. Ϫ Ϫ 2 ͑ ͒ ෇ ͑ ϩ ͒ kt͑ ͒ Ј͑ ͒ ෇ 59. y ෇ e x cos x 60. y ෇ sec x (b) Let h x 1 x x and show that h x 0 . (c) Deduce that t͑x͒ ෇ ͑1 ϩ x͒k. x 61. y ෇ 62. y ෇ e x ln͑1 ϩ x͒ sin x 76. In Exercise 53 in Section 10.2 it was shown that the length of the ellipse x ෇ a sin ␪ , y ෇ b cos ␪ , where a Ͼ b Ͼ 0 , is

63–70 ␲͞2 Find the sum of the series. L ෇ 4a y s1 Ϫ e 2 sin2 ␪ d␪ 0 ϱ x 4n ϱ ͑Ϫ1͒n ␲ 2n 63. ͚ ͑Ϫ ͒n 64. ͚ 1 2n ෇ s 2 Ϫ 2 ͞ n෇0 n! n෇0 6 ͑2n͒! where e a b a is the eccentricity of the ellipse. Expand the integrand as a binomial series and use the result ϱ 3n ϱ 3n 65. ͚ ͑Ϫ1͒nϪ1 66. ͚ of Exercise 50 in Section 7.1 to express L as a series in powers n n 6 n෇1 n 5 n෇0 5 n! of the eccentricity up to the term in e .

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WRITING PROJECT HOW NEWTON DISCOVERED THE BINOMIAL SERIES 767

LABORATORY PROJECT CAS AN ELUSIVE LIMIT

This project deals with the function sin͑tan x͒ Ϫ tan͑sin x͒ f ͑x͒ ෇ arcsin͑arctan x͒ Ϫ arctan͑arcsin x͒

1. Use your computer algebra system to evaluate f͑x͒ for x ෇ 1, 0.1, 0.01, 0.001,and 0.0001 . Does it appear that fx has a limit as l 0 ? 2. Use the CAS to graph fx near ෇ 0 . Does it appear that fx has a limit as l 0 ?

3. Try to evaluate limx l 0 f͑x͒ with l’Hospital’s Rule, using the CAS to ﬁnd derivatives of the numerator and denominator. What do you discover? How many applications of l’Hospital’s Rule are required?

4. Evaluate limx l 0 f͑x͒ by using the CAS to ﬁnd sufﬁciently many terms in the Taylor series of the numerator and denominator. (Use the command taylor in Maple or Series in Mathematica.)

5. Use the limit command on your CAS to ﬁnd limx l 0 f͑x͒ directly. (Most computer algebra systems use the method of Problem 4 to compute limits.) 6. In view of the answers to Problems 4 and 5, how do you explain the results of Problems 1 and 2?

CAS Computer algebra system required

WRITING PROJECT HOW NEWTON DISCOVERED THE BINOMIAL SERIES

The Binomial Theorem, which gives the expansion of ͑a ϩ b͒k , was known to Chinese mathe- maticians many centuries before the time of Newton for the case where the exponent k is a positive integer. In 1665, when he was 22, Newton was the first to discover the infinite series expansion of ͑a ϩ b͒k when k is a fractional exponent (positive or negative). He didn’t publish his discovery, but he stated it and gave examples of how to use it in a letter (now called the epistola prior) dated June 13, 1676, that he sent to Henry Oldenburg, secretary of the Royal Soci- ety of London, to transmit to Leibniz. When Leibniz replied, he asked how Newton had discovered the binomial series. Newton wrote a second letter, the epistola posterior of October 24, 1676, in which he explained in great detail how he arrived at his discovery by a very indirect route. He was investigating the areas under the curves y ෇ ͑1 Ϫ x 2 ͒n͞2 from 0 to x for n ෇ 0 , 1, 2, 3, 4, . . . . These are easy to calculate if n is even. By observing patterns and interpolating, Newton was able to guess the answers for odd values of n. Then he realized he could get the same answers by expressing ͑1 Ϫ x 2 ͒n͞2 as an infinite series. Write a report on Newton’s discovery of the binomial series. Start by giving the statement of the binomial series in Newton’s notation (see the epistola prior on page 285 of [4] or page 402 of [2]). Explain why Newton’s version is equivalent to Theorem 17 on page 761. Then read Newton’s epistola posterior (page 287 in [4] or page 404 in [2]) and explain the patterns that Newton discovered in the areas under the curves y ෇ ͑1 Ϫ x 2 ͒n͞2. Show how he was able to guess the areas under the remaining curves and how he verified his answers. Finally, explain how these discoveries led to the binomial series. The books by Edwards [1] and Katz [3] contain commentaries on Newton’s letters. 1. C. H. Edwards, The Historical Development of the Calculus (New York: Springer-Verlag, 1979), pp. 178Ð187.

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768 CHAPTER 11 INFINITE SEQUENCES AND SERIES

2. John Fauvel and Jeremy Gray, eds., The History of Mathematics: A Reader (London: MacMillan Press, 1987). 3. Victor Katz, A History of Mathematics: An Introduction (New York: HarperCollins, 1993), pp. 463–466. 4. D. J. Struik, ed., A Sourcebook in Mathematics, 1200–1800 (Princeton, NJ: Princeton University Press, 1969).

11.11 Applications of Taylor Polynomials

In this section we explore two types of applications of Taylor polynomials. First we look at how they are used to approximate functions––computer scientists like them because polynomials are the simplest of functions. Then we investigate how physicists and engineers use them in such ﬁelds as relativity, optics, blackbody radiation, electric dipoles, the velocity of water waves, and building highways across a desert.

Approximating Functions by Polynomials Suppose that f ͑x͒ is equal to the sum of its Taylor series at a:

ϱ f ͑n͒͑a͒ f ͑x͒ ෇ ͚ ͑x Ϫ a͒n n෇0 n!

In Section 11.10 we introduced the notation Tn͑x͒ for the n th partial sum of this series and called it the n th-degree Taylor polynomial of fa at . Thus

n ͑i͒͑ ͒ f a i Tn͑x͒ ෇ ͚ ͑x Ϫ a͒ i෇0 i!

f Ј͑a͒ f Љ͑a͒ f ͑n͒͑a͒ ෇ f ͑a͒ ϩ ͑x Ϫ a͒ ϩ ͑x Ϫ a͒2 ϩиииϩ ͑x Ϫ a͒n 1! 2! n!

Since fT is the sum of its Taylor series, we know that n͑x͒ l f ͑x͒ as n l ϱ and so Tn can be used as an approximation to ff : ͑x͒ϷTn͑x͒. Notice that the ﬁrst-degree Taylor polynomial

y T1͑x͒ ෇ f ͑a͒ ϩ f Ј͑a͒͑x Ϫ a͒ y=´ y=T£(x) y=T™(x) is the same as the linearization of f at a that we discussed in Section 3.10. Notice also that T1 and its derivative have the same values at a thatff and Ј have. In general, it can be y=T¡(x) shown that the derivatives of Tn at af agree with those of up to and including derivatives of order n . (0, 1) To illustrate these ideas let’s take another look at the graphs of y ෇ e x and its ﬁrst few x Taylor polynomials, as shown in Figure 1. The graph of T1 is the tangent line to y ෇ e 0 x at ͑0, 1͒ ; this tangent line is the best linear approximation to e x near ͑0, 1͒ . The graph 2 of T2 is the parabola y ෇ 1 ϩ x ϩ x ͞2, and the graph of T3 is the cubic curve 2 3 x y ෇ 1 ϩ x ϩ x ͞2 ϩ x ͞6, which is a closer ﬁt to the exponential curve y ෇ e than T2 . FIGURE 1 The next Taylor polynomial T4 would be an even better approximation, and so on.

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SECTION 11.11 APPLICATIONS OF TAYLOR POLYNOMIALS 769

෇ ෇ The values in the table give a numerical demonstration of the convergence of the Taylor x 0.2 x 3.0 x polynomials Tn͑x͒ to the function y ෇ e . We see that when x ෇ 0.2 the convergence is very T2͑x͒ 1.220000 8.500000 rapid, but when x ෇ 3 it is somewhat slower. In fact, the farther x is from 0, the more slowly ͑ ͒ x T4 x 1.221400 16.375000 Tn͑x͒ converges to e . ͑ ͒ T6 x 1.221403 19.412500 When using a Taylor polynomial Tn to approximate a functionf , we have to ask the ques- ͑ ͒ T8 x 1.221403 20.009152 tions: How good an approximation is it? How large should we take n to be in order ͑ ͒ T10 x 1.221403 20.079665 to achieve a desired accuracy? To answer these questions we need to look at the absolute e x 1.221403 20.085537 value of the remainder:

Խ Rn͑x͒ Խ ෇ Խ f ͑x͒ Ϫ Tn͑x͒ Խ

There are three possible methods for estimating the size of the error:

1. If a graphing device is available, we can use it to graph Խ Rn͑x͒ Խ and thereby estimate the error. 2. If the series happens to be an alternating series, we can use the Alternating Series Estimation Theorem. 3. In all cases we can use Taylor’s Inequality (Theorem 11.10.9), which says that if Խ f ͑nϩ1͒͑x͒ Խ ഛ M, then

M nϩ1 Խ Rn͑x͒ Խ ഛ Խ x Ϫ a Խ ͑n ϩ 1͒!

v EXAMPLE 1 (a) Approximate the function f ͑x͒ ෇ s3 x by a Taylor polynomial of degree 2 at a ෇ 8 . (b) How accurate is this approximation when 7 ഛ x ഛ 9 ?

SOLUTION (a) f ͑x͒ ෇ s3 x ෇ x 1͞3 f ͑8͒ ෇ 2

Ј͑ ͒ ෇ 1 Ϫ2͞3 Ј͑ ͒ ෇ 1 f x 3 x f 8 12

Љ͑ ͒ ෇ Ϫ 2 Ϫ5͞3 Љ͑ ͒ ෇ Ϫ 1 f x 9 x f 8 144

ٞ͑ ͒ ෇ 10 Ϫ8͞3 f x 27 x

Thus the second-degree Taylor polynomial is

f Ј͑8͒ f Љ͑8͒ T ͑x͒ ෇ f ͑8͒ ϩ ͑x Ϫ 8͒ ϩ ͑x Ϫ 8͒2 2 1! 2! ෇ ϩ 1 ͑ Ϫ ͒ Ϫ 1 ͑ Ϫ ͒2 2 12 x 8 288 x 8

The desired approximation is

s3 Ϸ ͑ ͒ ෇ ϩ 1 ͑ Ϫ ͒ Ϫ 1 ͑ Ϫ ͒2 x T2 x 2 12 x 8 288 x 8

(b) The Taylor series is not alternating when x Ͻ 8 , so we can’t use the Alternating Series Estimation Theorem in this example. But we can use Taylor’s Inequality with n ෇ 2 and a ෇ 8:

M 3 Խ R2͑x͒ Խ ഛ Խ x Ϫ 8 Խ 3!

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770 CHAPTER 11 INFINITE SEQUENCES AND SERIES where Խ f ٞ͑x͒ Խ ഛ Mx. Because ജ 7 , we have x 8͞3 ജ 78͞3 and so

10 1 10 1 f ٞ͑x͒ ෇ ؒ ഛ ؒ Ͻ 0.0021 27 x 8͞3 27 78͞3 2.5 Therefore we can take M ෇ 0.0021. Also 7 ഛ x ഛ 9 , so Ϫ1 ഛ x Ϫ 8 ഛ 1 and Խ x Ϫ 8 Խ ഛ 1. Then Taylor’s Inequality gives T™

0.0021 3 0.0021 Խ R2͑x͒ Խ ഛ ؒ 1 ෇ Ͻ 0.0004 y=œ# x„ 3! 6

Thus, if 7 ഛ x ഛ 9 , the approximation in part (a) is accurate to within 0.0004 . 0 15 FIGURE 2 Let’s use a graphing device to check the calculation in Example 1. Figure 2 shows that 3 the graphs of y ෇ sx and y ෇ T2͑x͒ are very close to each other when x is near 8. Fig- 0.0003 ure 3 shows the graph of Խ R2͑x͒ Խ computed from the expression

3 Խ R2͑x͒ Խ ෇ Խ sx Ϫ T2͑x͒ Խ y=|R™(x)| We see from the graph that

Խ R2͑x͒ Խ Ͻ 0.0003 79 0 when 7 ഛ x ഛ 9 . Thus the error estimate from graphical methods is slightly better than the FIGURE 3 error estimate from Taylor’s Inequality in this case.

v EXAMPLE 2 (a) What is the maximum error possible in using the approximation

x 3 x 5 sin x Ϸ x Ϫ ϩ 3! 5!

when Ϫ0.3 ഛ x ഛ 0.3? Use this approximation to ﬁnd sin 12Њ correct to six decimal places. (b) For what values of x is this approximation accurate to within 0.00005 ?

SOLUTION (a) Notice that the Maclaurin series

x 3 x 5 x 7 sin x ෇ x Ϫ ϩ Ϫ ϩиии 3! 5! 7! is alternating for all nonzero values of x , and the successive terms decrease in size because Խ x Խ Ͻ 1 , so we can use the Alternating Series Estimation Theorem. The error in approximating sin x by the ﬁrst three terms of its Maclaurin series is at most

x 7 Խ x Խ7 Ϳ Ϳ ෇ 7! 5040

If Ϫ0.3 ഛ x ഛ 0.3, then Խ x Խ ഛ 0.3 , so the error is smaller than

͑0.3͒7 Ϸ 4.3 ϫ 10Ϫ8 5040

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SECTION 11.11 APPLICATIONS OF TAYLOR POLYNOMIALS 771

To ﬁnd sin 12Њ we ﬁrst convert to radian measure:

12␲ ␲ sin 12Њ ෇ sinͩ ͪ ෇ sinͩ ͪ 180 15 ␲ ␲ 3 1 ␲ 5 1 Ϸ Ϫ ͩ ͪ ϩ ͩ ͪ Ϸ 0.20791169 15 15 3! 15 5!

Thus, correct to six decimal places, sin 12Њ Ϸ 0.207912. (b) The error will be smaller than 0.00005 if Խ x Խ7 Ͻ 0.00005 5040 Solving this inequality for x , we get

Խ x Խ7 Ͻ 0.252 or Խ x Խ Ͻ ͑0.252͒1͞7 Ϸ 0.821

So the given approximation is accurate to within 0.00005 when Խ x Խ Ͻ 0.82 .

TEC Module 11.10/11.11 graphically What if we use Taylor’s Inequality to solve Example 2? Since f ͑7͒͑x͒ ෇ Ϫcos x, we have shows the remainders in Taylor polynomial Խ f ͑7͒͑x͒ Խ ഛ 1 and so approximations. 1 7 Խ R6͑x͒ Խ ഛ Խ x Խ 7! 4.3 ϫ 10–* So we get the same estimates as with the Alternating Series Estimation Theorem. What about graphical methods? Figure 4 shows the graph of y=|Rß(x)| ͑ ͒ ෇ Ϫ Ϫ 1 3 ϩ 1 5 Խ R6 x Խ Խ sin x (x 6 x 120 x ) Խ

Ϫ8 and we see from it that Խ R6͑x͒ Խ Ͻ 4.3 ϫ 10 when Խ x Խ ഛ 0.3 . This is the same estimate _0.3 0.3 that we obtained in Example 2. For part (b) we want Խ R6͑x͒ Խ Ͻ 0.00005, so we graph both 0 y ෇ Խ R6͑x͒ Խ and y ෇ 0.00005 in Figure 5. By placing the cursor on the right intersection FIGURE 4 point we ﬁnd that the inequality is satisﬁed when Խ x Խ Ͻ 0.82 . Again this is the same estimate that we obtained in the solution to Example 2. 0.00006 If we had been asked to approximate sin 72Њ instead of sin 12Њ in Example 2, it would y=0.00005 have been wise to use the Taylor polynomials at a ෇ ␲͞3 (instead of a ෇ 0 ) because they are better approximations to sin xx for values of close to ␲͞3 . Notice that 72Њ is close to Њ ␲͞ ␲͞ y=|Rß(x)| 60 (or 3 radians) and the derivatives of sin x are easy to compute at 3 . Figure 6 shows the graphs of the Maclaurin polynomial approximations x 3 T1͑x͒ ෇ xT3͑x͒ ෇ x Ϫ _1 1 3! 0 x 3 x 5 x 3 x 5 x 7 FIGURE 5 T ͑x͒ ෇ x Ϫ ϩ T ͑x͒ ෇ x Ϫ ϩ Ϫ 5 3! 5! 7 3! 5! 7!

to the sine curve. You can see that as nT increases, n͑x͒ is a good approximation to sin x on a larger and larger interval. y T¡ T∞

0 x y=sin x FIGURE 6 T£ T¶

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772 CHAPTER 11 INFINITE SEQUENCES AND SERIES

One use of the type of calculation done in Examples 1 and 2 occurs in calculators and computers. For instance, when you press the sin or e x key on your calculator, or when a computer programmer uses a subroutine for a trigonometric or exponential or Bessel function, in many machines a polynomial approximation is calculated. The polynomial is often a Taylor polynomial that has been modiﬁed so that the error is spread more evenly through- out an interval.

Applications to Physics Taylor polynomials are also used frequently in physics. In order to gain insight into an equation, a physicist often simpliﬁes a function by considering only the ﬁrst two or three terms in its Taylor series. In other words, the physicist uses a Taylor polynomial as an approximation to the function. Taylor’s Inequality can then be used to gauge the accuracy of the approximation. The following example shows one way in which this idea is used in special relativity.

v EXAMPLE 3 In Einstein’s theory of special relativity the mass of an object moving with velocity v is m m ෇ 0 s1 Ϫ v 2͞c 2

where m0 is the mass of the object when at rest and c is the speed of light. The kinetic energy of the object is the difference between its total energy and its energy at rest:

2 2 K ෇ mc Ϫ m0c

(a) Show that when v is very small compared with cK , this expression for agrees with ෇ 1 2 classical Newtonian physics: K 2 m0v . (b) Use Taylor’s Inequality to estimate the difference in these expressions for K when Խ v Խ ഛ 100 m͞s.

SOLUTION (a) Using the expressions given for Km and , we get

The upper curve in Figure 7 is the graph of m c2 v2 Ϫ1͞2 the expression for the kinetic energy K of an ෇ 2 Ϫ 2 ෇ 0 Ϫ 2 ෇ 2ͫͩ Ϫ ͪ Ϫ ͬ K mc m0 c m0c m0 c 1 2 1 object with velocity v in special relativity. The s1 Ϫ v 2͞c2 c lower curve shows the function used for K in classical Newtonian physics. When is much v ෇ Ϫ 2͞ 2 ͑ ϩ ͒Ϫ1͞2 smaller than the speed of light, the curves are With x v c , the Maclaurin series for 1 x is most easily computed as a ෇ Ϫ1 Ͻ Ͻ practically identical. binomial series with k 2 . (Notice that Խ x Խ 1 because v c .) Therefore we have

K Ϫ 1 Ϫ 3 Ϫ 1 Ϫ 3 Ϫ 5 Ϫ ͞ ( )( ) ( )( )( ) ͑1 ϩ x͒ 1 2 ෇ 1 Ϫ 1 x ϩ 2 2 x 2 ϩ 2 2 2 x 3 ϩиии 2 2! 3!

K=mc@-m¸c@ ෇ Ϫ 1 ϩ 3 2 Ϫ 5 3 ϩиии 1 2 x 8 x 16 x

1 v2 3 v4 5 v6 and K ෇ m c2ͫͩ1 ϩ ϩ ϩ ϩиииͪ Ϫ 1ͬ 0 2 c2 8 c 4 16 c 6 K= 1 m¸√@ 2 1 v2 3 v4 5 v6 ෇ 2ͩ ϩ ϩ ϩиииͪ m0 c 0 c √ 2 c2 8 c 4 16 c 6

FIGURE 7 If v is much smaller than c , then all terms after the ﬁrst are very small when compared

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SECTION 11.11 APPLICATIONS OF TAYLOR POLYNOMIALS 773

with the ﬁrst term. If we omit them, we get

1 v2 K Ϸ m c2ͩ ͪ ෇ 1 m v2 0 2 c2 2 0

2 2 2 Ϫ1͞2 (b) If x ෇ Ϫv ͞c , f ͑x͒ ෇ m0 c ͓͑1 ϩ x͒ Ϫ 1͔, and M is a number such that Խ f Љ͑x͒ Խ ഛ M, then we can use Taylor’s Inequality to write

M 2 Խ R1͑x͒ Խ ഛ x 2! Љ͑ ͒ ෇ 3 2͑ ϩ ͒Ϫ5͞2 ഛ ͞ We have f x 4 m0 c 1 x and we are given that Խ v Խ 100 m s, so

3m c2 3m c2 Խ f Љ͑x͒ Խ ෇ 0 ഛ 0 ͑෇ M͒ 4͑1 Ϫ v 2͞c2 ͒5͞2 4͑1 Ϫ 1002͞c2 ͒5͞2

Thus, with c ෇ 3 ϫ 108 m͞s,

2 4 1 3m0 c 100 Ϫ10 Խ R1͑x͒ Խ ഛ ؒ ؒ Ͻ ͑4.17 ϫ 10 ͒m0 2 4͑1 Ϫ 1002͞c2 ͒5͞2 c 4

So when Խ v Խ ഛ 100 m͞s, the magnitude of the error in using the Newtonian expression Ϫ10 for kinetic energy is at most ͑4.2 ϫ 10 ͒m0.

Another application to physics occurs in optics. Figure 8 is adapted from Optics, 4th ed., by Eugene Hecht (San Francisco, 2002), page 153. It depicts a wave from the point source S meeting a spherical interface of radius R centered at C. The ray SA is refracted toward P.

¨r A ¨i

h Lo Li R ¨t V ˙ S CP

so si FIGURE 8 n¡ n™

Refraction at a spherical interface Courtesy of Eugene Hecht

Using Fermat’s principle that light travels so as to minimize the time taken, Hecht derives the equation n n 1 n s n s 1 1 ϩ 2 ෇ ͩ 2 i Ϫ 1 o ͪ ᐉo ᐉi R ᐉi ᐉo

where n1 and n2 are indexes of refraction and ᐉo , ᐉi , so , and si are the distances indicated in Figure 8. By the Law of Cosines, applied to triangles ACS and ACP, we have

2 2 2 ᐉo ෇ sR ϩ ͑so ϩ R͒ Ϫ 2R͑so ϩ R͒ cos ␾ Here we use the identity

2 2 cos͑␲ Ϫ ␾͒ ෇ Ϫcos ␾ ᐉi ෇ sR ϩ ͑si Ϫ R͒ ϩ 2R͑si Ϫ R͒ cos ␾

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774 CHAPTER 11 INFINITE SEQUENCES AND SERIES

Because Equation 1 is cumbersome to work with, Gauss, in 1841, simpliﬁed it by using the linear approximation cos ␾ Ϸ 1 for small values of ␾ . (This amounts to using the Taylor polynomial of degree 1.) Then Equation 1 becomes the following simpler equation [as you are asked to show in Exercise 34(a)]:

n n n Ϫ n 3 1 ϩ 2 ෇ 2 1 so si R

The resulting optical theory is known as Gaussian optics, or ﬁrst-order optics, and has become the basic theoretical tool used to design lenses. A more accurate theory is obtained by approximating cos ␾ by its Taylor polynomial of degree 3 (which is the same as the Taylor polynomial of degree 2). This takes into account rays for which ␾ is not so small, that is, rays that strike the surface at greater distances h above the axis. In Exercise 34(b) you are asked to use this approximation to derive the more accurate equation

n n n Ϫ n n 1 1 2 n 1 1 2 4 1 ϩ 2 ෇ 2 1 ϩ h 2ͫ 1 ͩ ϩ ͪ ϩ 2 ͩ Ϫ ͪͬ so si R 2so so R 2si R si

The resulting optical theory is known as third-order optics. Other applications of Taylor polynomials to physics and engineering are explored in Exercises 32, 33, 35, 36, 37, and 38, and in the Applied Project on page 777.

11.11 Exercises

8. ͑ ͒ ෇ ෇ ; 1. (a) Find the Taylor polynomials up to degree 6 for f x x cos xa, 0 ͑ ͒ ෇ ෇ Ϫ f x cos xacentered at 0 . Graph f and these 9. f ͑x͒ ෇ xe 2x, a ෇ 0 polynomials on a common screen. (b) Evaluate fx and these polynomials at ෇ ␲͞4 , ␲͞2 , 10. f ͑x͒ ෇ tanϪ1xa, ෇ 1 and ␲. (c) Comment on how the Taylor polynomials converge to f ͑x͒. CAS 11–12 Use a computer algebra system to ﬁnd the Taylor polynomials T centered at an for ෇ 2, 3, 4, 5. Then graph these ; 2. (a) Find the Taylor polynomials up to degree 3 for n polynomials and f on the same screen. f ͑x͒ ෇ 1͞xacentered at ෇ 1 . Graph f and these polynomials on a common screen. 11. f ͑x͒ ෇ cot xa, ෇ ␲͞4 (b) Evaluate fx and these polynomials at ෇ 0.9 and 1.3. 12. f ͑x͒ ෇ s3 1 ϩ x 2 , a ෇ 0 (c) Comment on how the Taylor polynomials converge to f ͑x͒. 13–22 ; 3–10 Find the Taylor polynomial T3͑x͒ for the function f (a) Approximate fn by a Taylor polynomial with degree at the centered at the number a . Graph fT and 3 on the same screen. number a. 3. ͑ ͒ ෇ ͞ ෇ f x 1 xa, 2 (b) Use Taylor’s Inequality to estimate the accuracy of the ͑ ͒Ϸ ͑ ͒ 4. f ͑x͒ ෇ x ϩ eϪx, a ෇ 0 approx i ma tion f x Tn x when x lies in the given interval. 5. f ͑x͒ ෇ cos xa, ෇ ␲͞2 ; (c) Check your result in part (b) by graphing Խ Rn͑x͒ Խ . 6. f ͑x͒ ෇ eϪx sin x, a ෇ 0 13. f ͑x͒ ෇ sx , a ෇ 4 , n ෇ 24 , ഛ x ഛ 4.2

7. f ͑x͒ ෇ ln xa, ෇ 1 14. f ͑x͒ ෇ xϪ2, a ෇ 1 , n ෇ 2 , 0.9 ഛ x ഛ 1.1

; Graphing calculator or computer required CAS Computer algebra system required 1. Homework Hints available at stewartcalculus.com

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CHAPTER 11.11 APPLICATIONS OF TAYLOR POLYNOMIALS 775

2͞3 15. f ͑x͒ ෇ x , a ෇ 1 , n ෇ 3 , 0.8 ഛ x ഛ 1.2 32. The resistivity ␳ of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters (⍀ -m). 16. f ͑x͒ ෇ sin xa, ෇ ␲͞6 , n ෇ 40 , ഛ x ഛ ␲͞3 The resistivity of a given metal depends on the temperature 17. f ͑x͒ ෇ sec xa, ෇ 0 , n ෇ 2 , Ϫ0.2 ഛ x ഛ 0.2 according to the equation

18. ͑ ͒ ෇ ͑ ϩ ͒ ෇ ෇ ഛ ഛ ␣͑tϪ20͒ f x ln 1 2x , a 1 , n 3 , 0.5 x 1.5 ␳͑t͒ ෇ ␳ 20 e

19. f ͑x͒ ෇ ex 2, a ෇ 0 , n ෇ 30 , ഛ x ഛ 0.1 where t is the temperature in ЊC . There are tables that list the ␣ ␳ 20. f ͑x͒ ෇ x ln xa, ෇ 1 , n ෇ 3 , 0.5 ഛ x ഛ 1.5 values of (called the temperature coefficient) and 20 (the resistivity at 20Њ C) for various metals. Except at very low 21. f ͑x͒ ෇ x sin xa, ෇ 0 , n ෇ 4 , Ϫ1 ഛ x ഛ 1 temperatures, the resis tivity varies almost linearly with temperature and so it is common to approximate the expression 22. ͑ ͒ ෇ ෇ ෇ Ϫ ഛ ഛ f x sinh 2xa, 0 , n 5 , 1 x 1 for ␳͑t͒ by its first- or second-degree Taylor polynomial at t ෇ 20. (a) Find expressions for these linear and quadratic 23. Њ Use the information from Exercise 5 to estimate cos 80 cor- approximations. rect to five decimal places. ; (b) For copper, the tables give ␣ ෇ 0.0039͞ЊC and ␳ ෇ ϫ Ϫ8 ⍀ 24. Use the information from Exercise 16 to estimate sin 38Њ 20 1.7 10 -m. Graph the resistivity of copper correct to five decimal places. and the linear and quadratic approximations for Ϫ250ЊഛC t ഛ 1000ЊC. 25. Use Taylor’s Inequality to determine the number of terms of ; (c) For what values of t does the linear approximation agree the Maclaurin series for e x that should be used to estimate with the exponential expression to within one percent? e 0.1 to within 0.00001 . 33. An electric dipole consists of two electric charges of equal 26. How many terms of the Maclaurin series for ln͑1 ϩ x͒ do magnitude and opposite sign. If the charges are q and Ϫq and you need to use to estimate ln 1.4 to within 0.001 ? are located at a distance d from each other, then the electric field EP at the point in the figure is ; 27–29 Use the Alternating Series Estimation Theorem or q q Taylor’s Inequality to estimate the range of values of x for which E ෇ Ϫ the given approximation is accurate to within the stated error. D2 ͑D ϩ d͒2 Check your answer graphically. By expanding this expression for E as a series in powers of x 3 27. sin x Ϸ x Ϫ (Խ error Խ Ͻ 0.01) d͞DE, show that is approximately proportional to 1͞D 3 6 when P is far away from the dipole. x 2 x 4 28. cos x Ϸ 1 Ϫ ϩ (Խ error Խ Ͻ 0.005) q _q 2 24 P D d x 3 x 5 29. arctan x Ϸ x Ϫ ϩ (Խ error Խ Ͻ 0.05) 3 5 34. (a) Derive Equation 3 for Gaussian optics from Equation 1 by approximating cos ␾ in Equation 2 by its first-degree 30. Suppose you know that Taylor polynomial. (b) Show that if cos ␾ is replaced by its third-degree Taylor ͑Ϫ1͒n n! polynomial in Equation 2, then Equation 1 becomes f ͑n͒͑4͒ ෇ 3n͑n ϩ 1͒ Equation 4 for third-order optics. [Hint: Use the first two Ϫ1 Ϫ1 terms in the binomial series for ᐉo and ᐉi . Also, use and the Taylor series of f centered at 4 converges to f ͑x͒ ␾ Ϸ sin ␾.] for all x in the interval of convergence. Show that the fifth- 35. If a water wave with length L moves with velocity v across a degree Taylor polynomial approximates f ͑5͒ with error less body of water with depth d , as in the figure on page 776, then than 0.0002. tL 2␲d 31. ͞ ͞ 2 v 2 ෇ tanh A car is moving with speed 20 m s and acceleration 2 m s 2␲ L at a given instant. Using a second-degree Taylor polyno mial, estimate how far the car moves in the next second. Would it (a) If the water is deep, show that v Ϸ stL͑͞2␲͒ . be reasonable to use this polynomial to estimate the distance (b) If the water is shallow, use the Maclaurin series for tanh traveled during the next minute? to show that v Ϸ std . (Thus in shallow water the veloc-

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776 CHAPTER 11 INFINITE SEQUENCES AND SERIES

ity of a wave tends to be independent of the length of the 38. The period of a pendulum with length L that makes a maxi -

wave.) mum angle ␪0 with the vertical is (c) Use the Alternating Series Estimation Theorem to show that if L Ͼ 10d , then the estimate v 2 Ϸ td is accurate to within L ␲͞2 dx T ෇ 4 ͱ y t 2 2 0.014 L. t 0 s1 Ϫ k sin x

෇ 1 ␪ t where k sin( 2 0 ) and is the acceleration due to gravity. (In L Exercise 42 in Section 7.7 we approximated this integral using d Simpson’s Rule.) (a) Expand the integrand as a binomial series and use the result of Exercise 50 in Section 7.1 to show that

36. A uniformly charged disk has radius R and surface charge den- L 12 1232 123252 T ෇ 2␲ͱ ͫ1 ϩ k 2 ϩ k 4 ϩ k 6 ϩиииͬ sity ␴ as in the ﬁgure. The electric potential VP at a point at a t 22 2242 224262 distance d along the perpendicular central axis of the disk is

If ␪0 is not too large, the approximation T Ϸ 2␲sL͞t , ෇ ␲ ␴ s 2 ϩ 2 Ϫ V 2 ke ( d R d) obtained by using only the ﬁrst term in the series, is often used. A better approximation is obtained by using two where ke is a constant (called Coulomb’s constant). Show that terms: ␲ 2␴ keR L V Ϸ for large d T Ϸ 2␲ͱ 1 ϩ 1 k 2 d t ( 4 )

(b) Notice that all the terms in the series after the ﬁrst one have R 1 coefﬁcients that are at most4 . Use this fact to compare this d series with a geometric series and show that P

L L 4 Ϫ 3k 2 2␲ͱ (1 ϩ 1 k 2 ) ഛ T ഛ 2␲ͱ t 4 t 4 Ϫ 4k 2

37. If a surveyor measures differences in elevation when making (c) Use the inequalities in part (b) to estimate the period of a ෇ ␪ ෇ Њ plans for a highway across a desert, corrections must be made pendulum with L 1 meter and 0 10 . How does it Ϸ ␲s ͞t for the curvature of the earth. compare with the estimate T 2 L ? What if ␪ ෇ Њ (a) If R is the radius of the earth and L is the length of the 0 42 ? highway, show that the correction is 39. In Section 4.8 we considered Newton’s method for approximating a root rf of the equation ͑x͒ ෇ 0 , and from an initial C ෇ R sec͑L͞R͒ Ϫ R approximation x1 we obtained successive approximations x , x , . . . , where (b) Use a Taylor polynomial to show that 2 3 ͑ ͒ ෇ Ϫ f xn L 2 5L 4 xnϩ1 xn Ј͑ ͒ C Ϸ ϩ f xn 2R 24R 3 Use Taylor’s Inequality with n ෇ 1 , a ෇ xn , and x ෇ r to show (c) Compare the corrections given by the formulas in parts (a) that if f Љ͑x͒ exists on an interval I containing rx , n , and xnϩ1 , and (b) for a highway that is 100 km long. (Take the radius and Խ f Љ͑x͒ Խ ഛ M, Խ f Ј͑x͒ Խ ജ Kxfor all ʦ I , then of the earth to be 6370 km.)

M 2 Խ xnϩ1 Ϫ r Խ ഛ Խ xn Ϫ r Խ L C 2K

R [This means that if xn is accurate to dx decimal places, then nϩ1 R is accurate to about 2d decimal places. More precisely, if the error at stage n is at most 10Ϫm , then the error at stage n ϩ 1 is at most ͑M͞2K͒10Ϫ2m.]

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APPLIED PROJECT RADIATION FROM THE STARS 777

APPLIED PROJECT RADIATION FROM THE STARS

Any object emits radiation when heated. A blackbody is a system that absorbs all the radiation that falls on it. For instance, a matte black surface or a large cavity with a small hole in its wall (like a blastfurnace) is a blackbody and emits blackbody radiation. Even the radiation from the sun is close to being blackbody radiation. Proposed in the late 19th century, the Rayleigh-Jeans Law expresses the energy density of blackbody radiation of wavelength ␭ as

8␲kT f ͑␭͒ ෇ ␭4

where ␭ is measured in meters, Tk is the temperature in kelvins (K), and is Boltzmann’s constant. The Rayleigh-Jeans Law agrees with experimental measurements for long wavelengths © Luke Dodd / Photo Researchers, Inc. but disagrees drastically for short wavelengths. [The law predicts that f ͑␭͒ l ϱ as ␭ l 0ϩ but experiments have shown that f ͑␭͒ l 0 .] This fact is known as the ultraviolet catastrophe. In 1900 Max Planck found a better model (known now as Planck’s Law) for blackbody radiation: 8␲hc␭Ϫ5 f ͑␭͒ ෇ e hc͑͞␭kT͒ Ϫ 1

where ␭ is measured in meters, T is the temperature (in kelvins), and

h ෇ Planck’s constant ෇ 6.6262 ϫ 10Ϫ34 Jиs

c ෇ speed of light ෇ 2.997925 ϫ 108 m͞s

k ෇ Boltzmann’s constant ෇ 1.3807 ϫ 10Ϫ23 J͞K

1. Use l’Hospital’s Rule to show that

lim f ͑␭͒ ෇ 0 and lim f ͑␭͒ ෇ 0 ␭ l 0ϩ ␭ l ϱ

for Planck’s Law. So this law models blackbody radiation better than the Rayleigh-Jeans Law for short wavelengths.

2. Use a Taylor polynomial to show that, for large wavelengths, Planck’s Law gives approximately the same values as the Rayleigh-Jeans Law.

; 3. Graph f as given by both laws on the same screen and comment on the similarities and differences. Use T ෇ 5700 K (the temperature of the sun). (You may want to change from meters to the more convenient unit of micrometers: 1 ␮m ෇ 10Ϫ6 m.)

4. Use your graph in Problem 3 to estimate the value of ␭ for which f ͑␭͒ is a maximum under Planck’s Law.

; 5. Investigate how the graph of fT changes as varies. (Use Planck’s Law.) In particular, graph fT for the stars Betelgeuse (෇ 3400 K ), Procyon (T ෇ 6400 K ), and Sirius (T ෇ 9200 K ), as well as the sun. How does the total radiation emitted (the area under the curve) vary with T ? Use the graph to comment on why Sirius is known as a blue star and Betelgeuse as a red star.

; Graphing calculator or computer required

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778 CHAPTER 11 INFINITE SEQUENCES AND SERIES

11 Review

Concept Check 1. (a) What is a convergent sequence? (c) If a series is convergent by the Alternating Series Test, how (b) What is a convergent series? do you estimate its sum?

(c) What does limn l ϱ an ෇ 3 mean? ͸ϱ (d) What does n෇1 an ෇ 3 mean? 8. (a) Write the general form of a power series. 2. (a) What is a bounded sequence? (b) What is the radius of convergence of a power series? (b) What is a monotonic sequence? (c) What is the interval of convergence of a power series? (c) What can you say about a bounded monotonic sequence? 9. Suppose f ͑x͒ is the sum of a power series with radius of 3. (a) What is a geometric series? Under what circumstances is convergence R. it convergent? What is its sum? (a) How do you differentiate f ? What is the radius of conver- (b) What is a p -series? Under what circumstances is it gence of the series for f Ј ? convergent? (b) How do you integrate f ? What is the radius of convergence ͸ x ͑ ͒ 4. Suppose an ෇ 3 and sn is the nth partial sum of the series. of the series for f x dx ?

What is limn l ϱ an ? What is limn l ϱ sn ? 10. (a) Write an expression for the nth -degree Taylor polyno mial 5. State the following. of fa centered at . (a) The Test for Divergence (b) Write an expression for the Taylor series of fa centered at . (b) The Integral Test (c) Write an expression for the Maclaurin series of f . (c) The Comparison Test (d) How do you show that f ͑x͒ is equal to the sum of its (d) The Limit Comparison Test Taylor series? (e) The Alternating Series Test (e) State Taylor’s Inequality. (f) The Ratio Test (g) The Root Test 11. Write the Maclaurin series and the interval of convergence for 6. (a) What is an absolutely convergent series? each of the following functions. (b) What can you say about such a series? (a) 1͑͞1 Ϫ x͒ (b) e x (c) What is a conditionally convergent series? (c) sin x (d) cos x Ϫ 7. (a) If a series is convergent by the Integral Test, how do you (e) tan 1x (f) ln͑1 ϩ x͒ estimate its sum? (b) If a series is convergent by the Comparison Test, how do 12. Write the binomial series expansion of ͑1 ϩ x͒k . What is the you estimate its sum? radius of convergence of this series?

True-False Quiz Determine whether the statement is true or false. If it is true, explain why. ϱ ͑Ϫ1͒n 1 If it is false, explain why or give an example that disproves the statement. 10. ͚ ෇ n෇0 n! e ͸ 1. If limn l ϱ an ෇ 0, then an is convergent. n 11. If Ϫ1 Ͻ ␣ Ͻ 1, then limn l ϱ ␣ ෇ 0. ϱ Ϫsin 1 2. The series ͸ ෇ n is convergent. n 1 ͸ ͸ 12. If an is divergent, then Խ an Խ is divergent. 3. If limn l ϱ an ෇ L, then limn l ϱ a2nϩ1 ෇ L. 4. ͸ n ͸ ͑Ϫ ͒n 13. ͑ ͒ ෇ Ϫ 2 ϩ 1 3 Ϫиии If cn6 is convergent, then cn 2 is convergent. If f x 2x x 3 x converges for all x , ٞ͑ ͒ ෇ ͸ n ͸ n then f 0 2. 5. If cn6 is convergent, then cn͑Ϫ6͒ is convergent. ͸ n 14. ͕ ͖͕͖͕ϩ ͖ 6. If cn x diverges when x ෇ 6 , then it diverges when x ෇ 10 . If an and bn are divergent, then an bn is divergent. 7. The Ratio Test can be used to determine whether ͸ 1͞n 3 15. If ͕an ͖͕ and bn ͖͕ are divergent, then an bn ͖ is divergent. converges. 16. ͕ ͖ Ͼ ͕ ͖ 8. The Ratio Test can be used to determine whether ͸ 1͞n! If an is decreasing and an 0 for all n , then an is converges. convergent. ͸ ͸ ͸ ͸ n 9. If 0 ഛ an ഛ bn and bn diverges, then an diverges. 17. If an Ͼ 0 and an converges, then ͑Ϫ1͒ an converges.

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CHAPTER 11 REVIEW 779

21. 18. If an Ͼ 0 and limn l ϱ ͑anϩ1͞an͒ Ͻ 1, then limn l ϱ an ෇ 0. If a ﬁnite number of terms are added to a convergent series, then the new series is still convergent. 19. 0.99999...෇ 1 ϱ ϱ ϱ 22. ෇ ෇ ෇ 20. If lim an ෇ 2 , then lim ͑anϩ3 Ϫ an͒ ෇ 0. If ͚ an A and ͚ bn B , then ͚ an bn AB. n l ϱ n l ϱ n෇1 n෇1 n෇1

Exercises 1–8 Determine whether the sequence is convergent or divergent. ϱ ͑Ϫ1͒n͑n ϩ 1͒3n ϱ ͑Ϫ1͒nsn 25. ͚ 26. ͚ If it is convergent, ﬁnd its limit. 2nϩ1 n෇1 2 n෇2 ln n 2 ϩ n3 9nϩ1 1. a ෇ 2. a ෇ n 1 ϩ 2n3 n 10n 27–31 Find the sum of the series. 3 n ϱ nϪ1 ϱ 3. ෇ 4. ෇ ͑ ␲͞ ͒ ͑Ϫ3͒ 1 an 2 an cos n 2 27. ͚ 28. ͚ 1 ϩ n 3n n෇1 2 n෇1 n͑n ϩ 3͒ n sin n ln n ϱ ϱ ͑Ϫ1͒n␲ n 5. a ෇ 6. a ෇ 29. ͚ ͓tanϪ1͑n ϩ 1͒ Ϫ tanϪ1n͔ 30. ͚ n 2 n 2n͑ ͒ n ϩ 1 sn n෇1 n෇0 3 2n ! e 2 e 3 e 4 7. ͕͑1 ϩ 3͞n͒4n ͖ 8. ͕͑Ϫ10͒n͞n!͖ 31. 1 Ϫ e ϩ Ϫ ϩ Ϫиии 2! 3! 4!

9. ෇ A sequence is defined recursively by the equations a1 1 , 32. Express the repeating decimal 4.17326326326...as a ෇ 1 ͑ ϩ ͕͖͒ Ͻ anϩ1 3 an 4 . Show that an is increasing and an 2 fraction. for all n . Deduce that ͕an ͖ is convergent and find its limit. 33. ജ ϩ 1 2 Show that cosh x 1 2 x for all x . 10. 4 Ϫn ෇ ; Show that lim n l ϱ n e 0 and use a graph to find the ϱ n 34. For what values of x does the series ͸ ෇ ͑ln x͒ converge? smallest value of N that corresponds to ␧ ෇ 0.1 in the pre- n 1 cise definition of a limit. ϱ ͑Ϫ1͒nϩ1 35. Find the sum of the series ͚ 5 correct to four deci- n෇1 n 11–22 Determine whether the series is convergent or divergent. mal places. ϱ ϱ ϱ 36. ͸ ͞ 6 n n2 ϩ 1 (a) Find the partial sum s5 of the series n෇1 1 n and 11. 12. ͚ 3 ͚ 3 estimate the error in using it as an approximation to the ෇ n ϩ 1 ෇ n ϩ 1 n 1 n 1 sum of the series. ϱ n3 ϱ ͑Ϫ1͒n (b) Find the sum of this series correct to five decimal places. 13. 14. ͚ n ͚ n෇1 5 n෇1 sn ϩ 1 37. Use the sum of the first eight terms to approximate the sum ͸ϱ n Ϫ1 of the series n෇1 ͑2 ϩ 5 ͒ . Estimate the error involved in ϱ 1 ϱ n 15. ͚ 16. ͚ lnͩ ͪ this approximation. n෇2 nsln n n෇1 3n ϩ 1 ϱ n n 38. (a) Show that the series ͚ is convergent. ϱ ϱ 2n ͑ ͒ cos 3n n n෇1 2n ! 17. 18. ͚ n ͚ 2 n n ෇ 1 ϩ ͑1.2͒ ෇ ͑1 ϩ 2n ͒ n n 1 n 1 (b) Deduce that lim ෇ 0. n l ϱ ͑2n͒! ϱ 1 ؒ 3 ؒ 5 ؒ иии ؒ ͑2n Ϫ 1͒ ϱ ͑Ϫ5͒2n 19. 20. ͸ϱ ͚ n ͚ 2 n 39. Prove that if the series n෇1 an is absolutely convergent, then n෇1 5 n! n෇1 n 9 the series ϱ ϱ ϩ Ϫ Ϫ ϱ Ϫ sn sn 1 sn 1 n ϩ 1 21. ͚ ͑Ϫ1͒n 1 22. ͚ ͩ ͪ ϩ ͚ an n෇1 n 1 n෇1 n n෇1 n is also absolutely convergent.

23–26 Determine whether the series is conditionally conver- 40–43 Find the radius of convergence and interval of convergent, absolutely convergent, or divergent. gence of the series. ϱ ϱ ϱ x n ϱ ͑x ϩ 2͒n 23. ͑Ϫ ͒nϪ1 Ϫ1͞3 24. ͑Ϫ ͒nϪ1 Ϫ3 40. ͑Ϫ ͒n 41. ͚ 1 n ͚ 1 n ͚ 1 2 n ͚ n n෇1 n෇1 n෇1 n 5 n෇1 n 4

; Graphing calculator or computer required

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780 CHAPTER 11 INFINITE SEQUENCES AND SERIES

ϱ ϱ 2n͑x Ϫ 2͒n 2n͑x Ϫ 3͒n ͑ ͒ 42. ͚ 43. ͚ ; (d) Check your result in part (c) by graphing Խ Rn x Խ . n෇1 ͑n ϩ 2͒! n෇0 sn ϩ 3 57. f ͑x͒ ෇ sx , a ෇ 1 , n ෇ 3 , 0.9 ഛ x ഛ 1.1

44. Find the radius of convergence of the series 58. f ͑x͒ ෇ sec x, a ෇ 0 , n ෇ 2 , 0 ഛ x ഛ ␲͞6 ϱ ͑ ͒ 2n ! n ͚ 2 x ෇ ͑n!͒ n 1 59. Use series to evaluate the following limit. 45. Find the Taylor series of f ͑x͒ ෇ sin x at a ෇ ␲͞6 . sin x Ϫ x 46. Find the Taylor series of f ͑x͒ ෇ cos x at a ෇ ␲͞3 . lim 3 x l 0 x 47–54 Find the Maclaurin series for f and its radius of conver- 60. The force due to gravity on an object with mass m at a gence. You may use either the direct method (definition of a height h above the surface of the earth is Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for x , sin xe , tanϪ1x , mtR2 and ln͑1 ϩ x͒. F ෇ ͑R ϩ h͒2 x 2 47. f ͑x͒ ෇ 48. f ͑x͒ ෇ tanϪ1͑x 2 ͒ 1 ϩ x where R is the radius of the earth and t is the acceleration due to gravity. 49. ͑ ͒ ෇ ͑ Ϫ ͒ 50. ͑ ͒ ෇ 2x f x ln 4 x f x xe (a) Express Fh as a series in powers of ͞R . ; (b) Observe that if we approximate F by the first term in the 51. ͑ ͒ ෇ ͑ 4 ͒ 52. ͑ ͒ ෇ x f x sin x f x 10 series, we get the expression F Ϸ mt that is usually used when hR is much smaller than . Use the Alter nating 4 Ϫ5 53. f ͑x͒ ෇ 1͞s16 Ϫ x 54. f ͑x͒ ෇ ͑1 Ϫ 3x͒ Series Estimation Theorem to estimate the range of values of hF for which the approximation Ϸ mt is accurate to within one percent. (Use R ෇ 6400 km.) e x 55. Evaluate y dx as an infinite series. ͸ϱ n x 61. Suppose that f ͑x͒ ෇ n෇0 cn x for all x . (a) If f is an odd function, show that 56. x1 s ϩ 4 Use series to approximate 0 1 x dx correct to two decimal places. c0 ෇ c2 ෇ c4 ෇ иии෇ 0 57–58 (b) If f is an even function, show that (a) Approximate f by a Taylor polynomial with degree n at the number a. c1 ෇ c3 ෇ c5 ෇ иии෇ 0 ; (b) Graph f and Tn on a common screen. (c) Use Taylor’s Inequality to estimate the accuracy of the 2 ͑2n͒! approximation f ͑x͒ϷT ͑x͒ when x lies in the given interval. 62. If f ͑x͒ ෇ ex , show that f ͑2n͒͑0͒ ෇ . n n!

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Problems Plus

ϱ ͑x ϩ 2͒n Before you look at the solution of the example, EXAMPLE ͚ Find the sum of the series ͑ ϩ ͒ . cover it up and ﬁrst try to solve the problem n෇0 n 3 ! yourself. SOLUTION The problem-solving principle that is relevant here is recognizing something familiar. Does the given series look anything like a series that we already know? Well, it does have some ingredients in common with the Maclaurin series for the exponential function:

ϱ x n x 2 x 3 e x ෇ ͚ ෇ 1 ϩ x ϩ ϩ ϩиии n෇0 n! 2! 3!

We can make this series look more like our given series by replacing xx by ϩ 2 :

ϱ ͑x ϩ 2͒n ͑x ϩ 2͒2 ͑x ϩ 2͒3 e xϩ2 ෇ ͚ ෇ 1 ϩ ͑x ϩ 2͒ ϩ ϩ ϩиии n෇0 n! 2! 3!

But here the exponent in the numerator matches the number in the denominator whose factorial is taken. To make that happen in the given series, let’s multiply and divide by ͑x ϩ 2͒3 :

ϱ ͑ ϩ ͒n ϱ ͑ ϩ ͒nϩ3 x 2 ෇ 1 x 2 ͚ 3 ͚ n෇0 ͑n ϩ 3͒! ͑x ϩ 2͒ n෇0 ͑n ϩ 3͒!

͑x ϩ 2͒3 ͑x ϩ 2͒4 ෇ ͑x ϩ 2͒Ϫ3ͫ ϩ ϩиииͬ 3! 4!

We see that the series between brackets is just the series for e xϩ2 with the ﬁrst three terms missing. So

ϱ ͑x ϩ 2͒n ͑x ϩ 2͒2 ͚ ෇ ͑x ϩ 2͒Ϫ3ͫe xϩ2 Ϫ 1 Ϫ ͑x ϩ 2͒ Ϫ ͬ n෇0 ͑n ϩ 3͒! 2!

Problems 1. If f ͑x͒ ෇ sin͑x 3 ͒, ﬁnd f ͑15͒͑0͒ .

2. A function f is deﬁned by

2n Ϫ ͑ ͒ ෇ x 1 P¢ f x lim 2n 4 P£ n l ϱ x ϩ 1 2 Where is f continuous? P™ 3. 1 ෇ 1 Ϫ 8 1 (a) Show that tan 2 x cot 2 x 2 cot x. A 1 P¡ (b) Find the sum of the series

ϱ 1 x ͚ n tan n n෇1 2 2 P∞ 4. Let ͕Pn ͖ be a sequence of points determined as in the ﬁgure. Thus Խ AP1 Խ ෇ 1 , nϪ1 FIGURE FOR PROBLEM 4 Խ Pn Pnϩ1 Խ ෇ 2 , and angle APn Pnϩ1 is a right angle. Find limn l ϱ ЄPn APnϩ1.

781

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5. To construct the snowflake curve, start with an equilateral triangle with sides of length 1 . Step 1 in the construction is to divide each side into three equal parts, construct an equilateral triangle on the middle part, and then delete the middle part (see the figure). Step 2 is to repeat step 1 for each side of the resulting polygon. This process is repeated at each succeeding step. The snowflake curve is the curve that results from repeating this process indefinitely.

(a) Let sn , ln , and pn represent the number of sides, the length of a side, and the total length of the n th approximating curve (the curve obtained after step n of the construction),

respectively. Find formulas for sn , ln , and pn .

(b) Show that pn l ϱ as n l ϱ . (c) Sum an infinite series to find the area enclosed by the snowflake curve. Note: Parts (b) and (c) show that the snowflake curve is infinitely long but encloses only a finite 1 area. 6. Find the sum of the series 1 1 1 1 1 1 1 1 ϩ ϩ ϩ ϩ ϩ ϩ ϩ ϩиии 2 3 4 6 8 9 12 where the terms are the reciprocals of the positive integers whose only prime factors are 2s and 3s. 7. (a) Show that for xy Ϫ1 , x Ϫ y arctan x Ϫ arctan y ෇ arctan 2 1 ϩ xy if the left side lies between Ϫ␲͞2 and ␲͞2 . 120 Ϫ 1 ෇ ␲͞ (b) Show that arctan 119 arctan 239 4. (c) Deduce the following formula of John Machin (1680–1751): ␲ 4 arctan 1 Ϫ arctan 1 ෇ 5 239 4 (d) Use the Maclaurin series for arctan to show that Ͻ 1 Ͻ 3 0.1973955597 arctan 5 0.1973955616 (e) Show that Ͻ 1 Ͻ 0.004184075 arctan 239 0.004184077 (f) Deduce that, correct to seven decimal places, ␲ Ϸ 3.1415927. FIGURE FOR PROBLEM 5 Machin used this method in 1706 to find ␲ correct to 100 decimal places. Recently, with the aid of computers, the value of ␲ has been computed to increasingly greater accuracy. In 2009 T. Daisuke and his team computed the value of ␲ to more than two trillion decimal places! 8. (a) Prove a formula similar to the one in Problem 7(a) but involving arccot instead of arctan . ͸ϱ 2 (b) Find the sum of the series n෇0 arccot͑n ϩ n ϩ 1͒. ͸ϱ 3 n 9. Find the interval of convergence of n෇1 n x and find its sum.

10. If a0 ϩ a1 ϩ a2 ϩиииϩak ෇ 0, show that

lim (a0 sn ϩ a1 sn ϩ 1 ϩ a2 sn ϩ 2 ϩиииϩak sn ϩ k ) ෇ 0 n l ϱ If you don’t see how to prove this, try the problem-solving strategy of using analogy (see page 75). Try the special cases k ෇ 1 and k ෇ 2 ﬁrst. If you can see how to prove the assertion for these cases, then you will probably see how to prove it in general.

ϱ 1 11. ͩ Ϫ ͪ Find the sum of the series ͚ ln 1 2 . n෇2 n 782

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12. Suppose you have a large supply of books, all the same size, and you stack them at the edge 1 of a table, with each book extending farther beyond the edge of the table than the one 1 2 1 4 beneath it. Show that it is possible to do this so that the top book extends entirely beyond the 1 6 8 table. In fact, show that the top book can extend any distance at all beyond the edge of the table if the stack is high enough. Use the following method of stacking: The top book extends half its length beyond the second book. The second book extends a quarter of its length beyond the third. The third extends one-sixth of its length beyond the fourth, and so on. (Try FIGURE FOR PROBLEM 12 it yourself with a deck of cards.) Consider centers of mass. 13. If the curve y ෇ e Ϫx͞10 sin x, x ജ 0, is rotated about the x-axis , the resulting solid looks like an inﬁnite decreasing string of beads. (a) Find the exact volume of the nth bead. (Use either a table of integrals or a computer algebra system.) (b) Find the total volume of the beads. 14. If p Ͼ 1 , evaluate the expression 1 1 1 1 ϩ ϩ ϩ ϩиии 2 p 3p 4 p 1 1 1 1 Ϫ ϩ Ϫ ϩиии 2 p 3p 4 p

15. Suppose that circles of equal diameter are packed tightly in n rows inside an equilateral tri-

angle. (The ﬁgure illustrates the case ෇ 4 .) If An is the area of the triangle and An is the total area occupied by the n rows of circles, show that A ␲ lim n ෇ n l ϱ A 2s3

16. A sequence ͕an ͖ is deﬁned recursively by the equations

a0 ෇ a1 ෇ 1 n͑n Ϫ 1͒an ෇ ͑n Ϫ 1͒͑n Ϫ 2͒anϪ1 Ϫ ͑n Ϫ 3͒anϪ2 ͸ϱ Find the sum of the series n෇0 an . FIGURE FOR PROBLEM 15 17. Taking the value of x x at 0 to be 1 and integrating a series term by term, show that

ϱ nϪ1 1 ͑Ϫ1͒ x ෇ y x dx ͚ n 0 n෇1 n

P¡ P∞ P™ 18. Starting with the vertices P1͑0, 1͒ , P2͑1, 1͒ , P3͑1, 0͒ , P4͑0, 0͒ of a square, we construct further

points as shown in the ﬁgure: P5 is the midpoint of P1P2, P6 is the midpoint of P2P3, P7 is the midpoint of P P , and so on. The polygonal spiral path P P P P P P P ... approaches a P˜ 3 4 1 2 3 4 5 6 7 point P inside the square. ͑ ͒ 1 ϩ ϩ ϩ ෇ Pˆ (a) If the coordinates of Pn are xn, yn , show that 2 xn xnϩ1 xnϩ2 xnϩ3 2 and ﬁnd a Pß similar equation for the y -coordinates. (b) Find the coordinates of P .

P¡¸ ϱ ͑Ϫ1͒n 19. Find the sum of the series ͚ n . n෇1 ͑2n ϩ 1͒3 P¢ P¶ P£ 20. Carry out the following steps to show that FIGURE FOR PROBLEM 18 1 1 1 1 ϩ ϩ ϩ ϩиии෇ ln 2 8 ؒ 7 6 ؒ 5 4 ؒ 3 2 ؒ 1

(a) Use the formula for the sum of a ﬁnite geometric series (11.2.3) to get an expression for

1 Ϫ x ϩ x 2 Ϫ x 3 ϩиииϩx 2nϪ2 Ϫ x 2nϪ1

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(b) Integrate the result of part (a) from 0 to 1 to get an expression for

1 1 1 1 1 1 Ϫ ϩ Ϫ ϩиииϩ Ϫ 2 3 4 2n Ϫ 1 2n as an integral. (c) Deduce from part (b) that

1 1 1 1 1 dx 1 Ϳ ϩ ϩ ϩиииϩ Ϫ y Ϳ Ͻ y x 2n dx 2n Ϫ 1͒͑2n͒ 0 1 ϩ x 0͑ 6 ؒ 5 4 ؒ 3 2 ؒ 1

(d) Use part (c) to show that the sum of the given series is ln 2 . 21. Find all the solutions of the equation x x 2 x 3 x 4 1 ϩ ϩ ϩ ϩ ϩиии෇ 0 2! 4! 6! 8! Hint: Consider the cases x ജ 0 and x Ͻ 0 separately. 22. 1 1 Right-angled triangles are constructed as in the ﬁgure. Each triangle has height 1 and its base is the hypotenuse of the preceding triangle. Show that this sequence of triangles makes indef- ͸ 1 1 initely many turns around P by showing that ␪ n is a divergent series. 23. Consider the series whose terms are the reciprocals of the positive integers that can be written ¨£ in base 10 notation without using the digit 0. Show that this series is convergent and the sum ¨™ 1 is less than 90. ¨¡ 24. P 1 (a) Show that the Maclaurin series of the function x ϱ FIGURE FOR PROBLEM 22 ͑ ͒ ෇ n f x 2 is ͚ fn x 1 Ϫ x Ϫ x n෇1

where fn is the nth Fibonacci number, that is, f1 ෇ 1 , f2 ෇ 1 , and fn ෇ fnϪ1 ϩ fnϪ2 2 2 for n ജ 3 . [Hint: Write x͑͞1 Ϫ x Ϫ x ͒ ෇ c0 ϩ c1x ϩ c2 x ϩ . . . and multiply both sides of this equation by 1 Ϫ x Ϫ x 2 .] (b) By writing f ͑x͒ as a sum of partial fractions and thereby obtaining the Maclaurin series in a different way, ﬁnd an explicit formula for the nth Fibonacci number.

x 3 x 6 x 9 25. Let u ෇ 1 ϩ ϩ ϩ ϩиии 3! 6! 9! x 4 x 7 x 10 v ෇ x ϩ ϩ ϩ ϩиии 4! 7! 10! x 2 x 5 x 8 w ෇ ϩ ϩ ϩиии 2! 5! 8!

Show that u 3 ϩ v3 ϩ w3 Ϫ 3uvw ෇ 1. 26 Prove that if n Ͼ 1 , the nth partial sum of the harmonic series is not an integer. Hint: Let 2k be the largest power of 2 that is less than or equal to nM and let be the product

of all odd integers that are less than or equal to ns . Suppose that n ෇ m , an integer. Then k k M2 sn ෇ M2 m. The right side of this equation is even. Prove that the left side is odd by showing that each of its terms is an even integer, except for the last one.

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Vectors and the 12 Geometry of Space

Examples of the surfaces and solids we study in this chapter are paraboloids (used for satellite dishes) and hyper- boloids (used for cooling towers of nuclear reactors). © David Frazier / Corbis © Mark C. Burnett / Photo Researchers, Inc

In this chapter we introduce vectors and coordinate systems for three-dimensional space. This will be the setting for our study of the calculus of functions of two variables in Chapter 14 because the graph of such a function is a surface in space. In this chapter we will see that vectors provide particularly simple descriptions of lines and planes in space.

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