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Infinite Sequences and Series Were Introduced Briely in a Preview of Calculus in Con- Nection with Zeno’S Paradoxes and the Decimal Representation of Numbers

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Infinite Sequences and Series Were Introduced Briely in a Preview of Calculus in Con- Nection with Zeno’S Paradoxes and the Decimal Representation of Numbers 11 Infinite Sequences and Series Betelgeuse is a red supergiant star, one of the largest and brightest of the observable stars. In the project on page 783 you are asked to compare the radiation emitted by Betelgeuse with that of other stars. STScI / NASA / ESA / Galaxy / Galaxy Picture Library / Alamy INFINITE SEQUENCES AND SERIES WERE introduced briely in A Preview of Calculus in con- nection 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 ininite series. For instance, in inding areas he often integrated a function by irst expressing it as a series and then integrating each term of the series. We will pursue his idea in Section 11.10 in order to integrate such functions as e2x 2. (Recall that we have previously been unable to do this.) Many of the func- tions that arise in mathematical physics and chemistry, such as Bessel functions, are deined as sums of series, so it is important to be familiar with the basic concepts of convergence of ininite sequences and series. Physicists also use series in another way, as we will see in Section 11.11. In studying ields as diverse as optics, special relativity, and electromagnetism, they analyze phenomena by replacing a function with the irst few terms in the series that represents it. 693 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 694 CHAPTER 11 Ininite Sequences and Series A sequence can be thought of as a list of numbers written in a deinite order: a1, a2, a3, a4,..., an, . The number a1 is called the irst term, a2 is the second term, and in general an is the nth term. We will deal exclusively with ininite sequences and so each term an will have a successor an11. Notice that for every positive integer n there is a corresponding number an and so a sequence can be deined as a function whose domain is the set of positive integers. But we usually write an instead of the function notation f snd for the value of the function at the number n. NOTATION The sequence {a1, a2, a3, . .} is also denoted by ` han j or han j n−1 EXAMPLE 1 Some sequences can be deined 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 deining formula, and a third by writing out the terms of the sequence. Notice that n doesn’t have to start at 1. n ` n 1 2 3 4 n (a) an − , , , , . , , . .. H n 1 1J − n 1 1 H 2 3 4 5 n 1 1 J n 1 s21dnsn 1 1d s21dnsn 1 1d 2 3 4 5 s21dnsn 1 1d (b) a − 2 , , 2 , , . , , . H 3n J n 3n H 3 9 27 81 3n J ` − (c) hsn 2 3 jn−3 an sn 2 3 , n > 3 h0, 1, s2 , s3 , . , sn 2 3 , . .j ` n␲ n␲ s3 1 n␲ (d) cos an − cos , n > 0 1, , , 0, . , cos , . ■ H 6 J − 6 H 2 2 6 J n 0 EXAMPLE 2 Find a formula for the general term an of the sequence 3 4 5 6 7 , 2 , , 2 , , . H 5 25 125 625 3125 J assuming that the pattern of the irst few terms continues. SOLUTION We are given that 3 4 5 6 7 a − a − 2 a − a − 2 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 numer- ator 5; in general, the nth term will have numerator n 1 2. The denominators are the Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SECTION 11.1 Sequences 695 n powers of 5, so an has denominator 5 . The signs of the terms are alternately positive and negative, so we need to multiply by a power of 21. In Example 1(b) the factor s21d n meant we started with a negative term. Here we want to start with a positive term and so we use s21d n21 or s21d n11. Therefore n21 n 1 2 an − s21d ■ 5 n EXAMPLE 3 Here are some sequences that don’t have a simple deining equation. (a) The sequence hpn j, 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 han j is a well-deined sequence whose irst few terms are h7, 1, 8, 2, 8, 1, 8, 2, 8, 4, 5, . .j (c) The Fibonacci sequence h fn j is deined recursively by the conditions f1 − 1 f2 − 1 fn − fn21 1 fn22 n > 3 Each term is the sum of the two preceding terms. The irst few terms are h1, 1, 2, 3, 5, 8, 13, 21, . .j 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 − nysn 1 1d, can be pictured either a¡ a™ a£ by plotting its terms on a number line, as in Figure 1, or by plotting its graph, as in Fig- 011 ure 2. Note that, since a sequence is a function whose domain is the set of positive inte- 2 gers, its graph consists of isolated points with coordinates FIGURE 1 s1, a1d s2, a2 d s3, a3 d . sn, an d . an From Figure 1 or Figure 2 it appears that the terms of the sequence an − nysn 1 1d are approaching 1 as n becomes large. In fact, the difference 1 n 1 7 1 2 − a¶= 8 n 1 1 n 1 1 0 1 2 3 4 5 6 7 n can be made as small as we like by taking n suficiently large. We indicate this by writing n FIGURE 2 lim − 1 n l ` n 1 1 In general, the notation lim an − L n l ` means that the terms of the sequence han j approach L as n becomes large. Notice that the following deinition of the limit of a sequence is very similar to the deinition of a limit of a function at ininity given in Section 2.6. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 696 CHAPTER 11 Ininite Sequences and Series 1 Deinition A sequence han j has the limit L and we write lim an − L or an l L as n l ` n l ` if we can make the terms an as close to L as we like by taking n suficiently large. If limn l ` an exists, we say the sequence converges (or is convergent). Otherwise, we say the sequence diverges (or is divergent). Figure 3 illustrates Deinition 1 by showing the graphs of two sequences that have the limit L. an an L L FIGURE 3 Graphs of two sequences with lim an − L 0 n 0 n n l ` A more precise version of Deinition 1 is as follows. 2 Deinition A sequence han j has the limit L and we write lim an − L or an l L as n l ` Compare this deinition with n l ` Deinition 2.6.7. if for every «. 0 there is a corresponding integer N such that if n . N then | an 2 L | ,« Deinition 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 sL 2 «, L 1 «d is chosen, there exists an N such that all terms of the sequence from aN11 onward must lie in that interval. a¡ a£ a™aˆaN+1 aN+2 a˜ aß a∞ a¢ a¶ 0 L-∑ L L+∑ FIGURE 4 Another illustration of Deinition 2 is given in Figure 5. The points on the graph of han j must lie between the horizontal lines y − L 1 « and y − L 2 « if n . N. This picture must be valid no matter how small « is chosen, but usually a smaller « requires a larger N. y y=L+∑ L y=L-∑ n 0 3 4 FIGURE 5 1 2 N Copyright 2016 Cengage Learning.
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