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676 CHAPTER 9 Infinite

Section 9.10 Taylor and Maclaurin Series • Find a Taylor or Maclaurin series for a function. • Find a . • Use a basic list of to find other Taylor series.

Taylor Series and Maclaurin Series In Section 9.9, you derived power series for several functions using with term-by-term differentiation or integration. In this section you will study a general procedure for deriving the power series for a function that has of all orders. The following theorem gives the form that every convergent power series must take.

Bettmann/Corbis THEOREM 9.22 The Form of a Convergent Power Series COLIN MACLAURIN (1698–1746) n If f is represented by a power series f x an x c for all x in an open The development of power series to represent interval I containing c, then a f ncn! and functions is credited to the combined work of n many seventeenth and eighteenth century fc f nc fx fc fcx c x c2 . . . x cn . . . . mathematicians. Gregory, Newton, John and 2! n! James Bernoulli, Leibniz, Euler, Lagrange, Wallis, and Fourier all contributed to this work. However, the two names that are most Proof Suppose the power series a x cn has a radius of convergence R. Then, commonly associated with power series are n by Theorem 9.21, you know that the nth of f exists for x c < R, and Brook Taylor (1685–1731) and Colin Maclaurin. by successive differentiation you obtain the following. 0 2 3 4 . . . f x a0 a1 x c a2 x c a3 x c a4 x c 1 2 3 . . . f x a1 2a2 x c 3a3 x c 4a4 x c 2 2 . . . f x 2a2 3!a3 x c 4 3a4 x c 3 . . . f x 3!a3 4!a4 x c n . . . f x n!an n 1 !an1 x c Evaluating each of these derivatives at x c yields 0 f c 0!a0 1 f c 1!a1 2 f c 2!a2 3 f c 3!a3 n and, in general,f c n!an. By solving for an, you find that the coefficients of the power series representation of fx are NOTE Be sure you understand Theorem f n c a . 9.22. The theorem says that if a power n n! series converges to f x, the series must be a Taylor series. The theorem does not say that every series formed with the Notice that the coefficients of the power series in Theorem 9.22 are precisely the n Taylor coefficients an f c n! will coefficients of the Taylor polynomials for f x at c as defined in Section 9.7. For this converge to f x. reason, the series is called the Taylor series for fx at c. 332460_0910.qxd 11/4/04 3:12 PM Page 677

SECTION 9.10 Taylor and Maclaurin Series 677

Definitions of Taylor and Maclaurin Series If a function f has derivatives of all orders at x c, then the series f nc f nc x cn fc fcx c . . . x cn . . . n0 n! n! is called the Taylor series for fx at c. Moreover, if c 0, then the series is the Maclaurin series for f.

If you know the pattern for the coefficients of the Taylor polynomials for a function, you can extend the pattern easily to form the corresponding Taylor series. For instance, in Example 4 in Section 9.7, you found the fourth Taylor polynomial for ln x, centered at 1, to be 1 1 1 P x x 1 x 12 x 13 x 14. 4 2 3 4 From this pattern, you can obtain the Taylor series for ln x centered at c 1, 1 1n1 x 1 x 12 . . . x 1n . . . . 2 n

EXAMPLE 1 Forming a Power Series

Use the function fx sin x to form the Maclaurin series f n0 f0 f 30 f 40 xn f0 f0x x2 x3 x 4 . . . n0 n! 2! 3! 4! and determine the interval of convergence.

Solution Successive differentiation of fx yields fx sin x f0 sin 0 0 fx cos x f0 cos 0 1 fx sin x f0 sin 0 0 f 3x cos x f 30 cos 0 1 f 4x sin x f 40 sin 0 0 f 5x cos x f 50 cos 0 1 and so on. The pattern repeats after the third derivative. So, the power series is as follows. f n0 f0 f 30 f 40 xn f0 f0x x2 x3 x 4 . . . n0 n! 2! 3! 4! 1n x2n1 0 1 0 1 0 2 3 4 5 6 0 1 x x x x x x n0 2n 1 ! 2! 3! 4! 5! 6! 1 x7 . . . 7! x3 x5 x7 x . . . 3! 5! 7! By the , you can conclude that this series converges for all x. 332460_0910.qxd 11/4/04 3:12 PM Page 678

678 CHAPTER 9 Infinite Series

π −1, x < − Notice that in Example 1 you cannot conclude that the power series converges to 2 π sin x for all x. You can simply conclude that the power series converges to some f(x) = sin x, x ≤ 2 function, but you are not sure what function it is. This is a subtle, but important, point π 1, x > in dealing with Taylor or Maclaurin series. To persuade yourself that the series 2 y fc f nc fc fcx c x c2 . . . x cn . . . 2! n! 1 might converge to a function other than f, remember that the derivatives are being x evaluated at a single point. It can easily happen that another function will agree with −πππ the values of f nx when x c and disagree at other x- values. For instance, if you 22 formed the power series (centered at 0) for the function shown in Figure 9.23, you −1 would obtain the same series as in Example 1. You know that the series converges for all x, and yet it obviously cannot converge to both fx and sin x for all x. Figure 9.23 Let f have derivatives of all orders in an open interval I centered at c. The Taylor series for f may fail to converge for some x in I. Or, even if it is convergent, it may fail to have f x as its sum. Nevertheless, Theorem 9.19 tells us that for each n, fc f nc f x f c fcx c x c2 . . . x cn R x, 2! n! n where f n1z R x x cn1. n n 1! Note that in this remainder formula the particular value of z that makes the → remainder formula true depends on the values of x and n. If Rn 0, then the following theorem tells us that the Taylor series for f actually converges to f x for all x in I.

THEOREM 9.23 Convergence of Taylor Series If lim Rn 0 for all x in the interval I, then the Taylor series for f converges n→ and equals f x, f nc f x x cn. n0 n!

Proof For a Taylor series, the nth partial sum coincides with the nth Taylor polyno- mial. That is,Sn x Pn x . Moreover, because Pn x f x Rn x it follows that lim Sn x lim Pn x n→ n→ lim f x Rn x n→ f x lim Rn x . n→ So, for a given x, the Taylor series (the sequence of partial sums) converges to fx → → if and only if Rn x 0 as n .

NOTE Stated another way, Theorem 9.23 says that a power series formed with Taylor n coefficients an f c n! converges to the function from which it was derived at precisely those values for which the remainder approaches 0 as n → . 332460_0910.qxd 11/4/04 3:12 PM Page 679

SECTION 9.10 Taylor and Maclaurin Series 679

In Example 1, you derived the power series from the sine function and you also concluded that the series converges to some function on the entire real line. In Example 2, you will see that the series actually converges to sin x. The key observa- tion is that although the value of z is not known, it is possible to obtain an upper bound for f n1z.

EXAMPLE 2 A Convergent Maclaurin Series

Show that the Maclaurin series for fx sin x converges to sin x for all x.

Solution Using the result in Example 1, you need to show that x3 x5 x7 1n x2n1 sin x x . . . . . . 3! 5! 7! 2n 1! is true for all x. Because f n1x ±sin x or f n1x ±cos x you know that f n1z ≤ 1 for every real number z. Therefore, for any fixed x, you can apply Taylor’s Theorem (Theorem 9.19) to conclude that f n1z xn1 0 ≤ R x xn1 ≤ . n n 1! n 1! From the discussion in Section 9.1 regarding the relative rates of convergence of exponential and sequences, it follows that for a fixed x xn1 lim 0. n→ n 1! → → Finally, by the Squeeze Theorem, it follows that for all x, Rn x 0 as n . So, by Theorem 9.23, the Maclaurin series for sin x converges to sin x for all x.

Figure 9.24 visually illustrates the convergence of the Maclaurin series for sin x by comparing the graphs of the Maclaurin polynomials P1 x , P3 x , P5 x , and P7 x with the graph of the sine function. Notice that as the degree of the polynomial increases, its graph more closely resembles that of the sine function.

y y y y

4 4 4 4 y = sin x 3 3 y = sin x 3 3 2 2 2 2 1 1 1 1 x x x x −πππ 2 −π 2π ππ2 −π π 2π −2 −2 −2 −2 −3 y = sin x −3 −3 yx= sin −3 −4 −4 −4 −4 3 3 5 3 5 7 P (x) = x P (x) = x − x Px() = x− x + x P (x) = x − x + x − x 1 3 3! 5 3! 5! 7 3! 5! 7!

As n increases, the graph ofPn more closely resembles the sine function. Figure 9.24 332460_0910.qxd 11/4/04 3:12 PM Page 680

680 CHAPTER 9 Infinite Series

The guidelines for finding a Taylor series for f x at c are summarized below.

Guidelines for Finding a Taylor Series 1. Differentiate fx several times and evaluate each derivative at c. fc, fc, fc, fc, . . . , f n c, . . . Try to recognize a pattern in these numbers. 2. Use the sequence developed in the first step to form the Taylor coefficients n an f c n!, and determine the interval of convergence for the resulting power series f c f nc fc fcx c x c2 . . . x cn . . . . 2! n! 3. Within this interval of convergence, determine whether or not the series converges to fx.

The direct determination of Taylor or Maclaurin coefficients using successive differentiation can be difficult, and the next example illustrates a shortcut for finding the coefficients indirectly—using the coefficients of a known Taylor or Maclaurin series.

EXAMPLE 3 Maclaurin Series for a Composite Function

Find the Maclaurin series for fx sin x2.

Solution To find the coefficients for this Maclaurin series directly, you must calculate successive derivatives of fx sin x2. By calculating just the first two, fx 2x cos x2 and fx 4x2 sin x2 2 cos x2 you can see that this task would be quite cumbersome. Fortunately, there is an alternative. First consider the Maclaurin series for sin x found in Example 1. gx sin x x3 x5 x7 x . . . 3! 5! 7! Now, because sin x2 gx2, you can substitute x2 for x in the series for sin x to obtain sin x2 gx2 x 6 x10 x14 x2 . . . . 3! 5! 7!

Be sure to understand the point illustrated in Example 3. Because direct computation of Taylor or Maclaurin coefficients can be tedious, the most practical way to find a Taylor or Maclaurin series is to develop power series for a basic list of elementary functions. From this list, you can determine power series for other functions by the operations of addition, subtraction, multiplication, division, differen- tiation, integration, or composition with known power series. 332460_0910.qxd 11/4/04 3:12 PM Page 681

SECTION 9.10 Taylor and Maclaurin Series 681

Binomial Series Before presenting the basic list for elementary functions, you wll develop one more series—for a function of the form fx 1 xk. This produces the binomial series.

EXAMPLE 4 Binomial Series

Find the Maclaurin series for fx 1 xk and determine its radius of convergence. Assume that k is not a positive integer.

Solution By successive differentiation, you have fx 1 xk f0 1 fx k1 xk1 f0 k fx kk 11 xk2 f0 kk 1 fx kk 1k 21 xk3 f0 kk 1k 2 f nx k . . . k n 11 xkn f n0 kk 1 . . . k n 1 which produces the series kk 1x2 kk 1 . . . k n 1xn 1 kx . . . . . . . 2 n! → Because an1 an 1, you can apply the Ratio Test to conclude that the radius of convergence is R 1. So, the series converges to some function in the interval 1, 1.

Note that Example 4 shows that the Taylor series for 1 xk converges to some function in the interval 1, 1. However, the example does not show that the series k actually converges to 1 x . To do this, you could show that the remainder Rn x converges to 0, as illustrated in Example 2.

EXAMPLE 5 Finding a Binomial Series

Find the power series for fx 3 1 x.

Solution Using the binomial series kk 1x2 kk 1k 2x3 1 xk 1 kx . . . 2! 3! 1 let k 3 and write 2 3 4 x 2x 2 5x 2 5 8x 2 1 x1 3 1 . . . 3 322! 333! 344! which converges for 1 ≤ x ≤ 1.

P4

−22TECHNOLOGY Use a graphing utility to confirm the result in Example 5. When you graph the functions

2 3 4 − 13 x x 5x 10x 3 1 fx 1 x and P x 1 f(x) = 1 + x 4 3 9 81 243 Figure 9.25 in the same viewing window, you should obtain the result shown in Figure 9.25. 332460_0910.qxd 11/4/04 3:12 PM Page 682

682 CHAPTER 9 Infinite Series

Deriving Taylor Series from a Basic List The following list provides the power series for several elementary functions with the corresponding intervals of convergence.

Power Series for Elementary Functions Interval of Function Convergence 1 1 x 1 x 12 x 13 x 14 . . . 1n x 1n . . . 0 < x < 2 x

1 1 x x2 x3 x 4 x5 . . . 1n xn . . . 1 < x < 1 1 x

x 12 x 13 x 14 1n1x 1n ln x x 1 . . . . . . 0 < x ≤ 2 2 3 4 n

x2 x3 x 4 x5 xn ex 1 x . . . . . . < x < 2! 3! 4! 5! n!

x3 x5 x7 x9 1n x2n1 sin x x . . . . . . < x < 3! 5! 7! 9! 2n 1!

x2 x 4 x6 x8 1n x2n cos x 1 . . . . . . < x < 2! 4! 6! 8! 2n!

x3 x5 x7 x9 1n x2n1 arctan x x . . . . . . 1 ≤ x ≤ 1 3 5 7 9 2n 1

x3 1 3x5 1 3 5x7 2n!x2n1 arcsin x x . . . . . . 1 ≤ x ≤ 1 2 3 2 4 5 2 4 6 7 2nn!22n 1

kk 1x2 kk 1k 2x3 kk 1k 2k 3x 4 1 xk 1 kx . . . 1 < x < 1* 2! 3! 4!

* The convergence at x ±1 depends on the value of k.

NOTE The binomial series is valid for noninteger values of k. Moreover, if k happens to be a positive integer, the binomial series reduces to a simple binomial expansion.

EXAMPLE 6 Deriving a Power Series from a Basic List

Find the power series for fx cosx.

Solution Using the power series x2 x 4 x6 x8 cos x 1 . . . 2! 4! 6! 8! you can replace x by x to obtain the series x x2 x3 x4 cosx 1 . . . . 2! 4! 6! 8! This series converges for all x in the domain of cosx —that is, for x ≥ 0. 332460_0910.qxd 11/4/04 3:12 PM Page 683

SECTION 9.10 Taylor and Maclaurin Series 683

Power series can be multiplied and divided like polynomials. After finding the first few terms of the product (or quotient), you may be able to recognize a pattern.

EXAMPLE 7 Multiplication and Division of Power Series

Find the first three nonzero terms in each of the Maclaurin series. a.ex arctan x b. tan x

Solution a. Using the Maclaurin series for ex and arctan x in the table, you have x x2 x3 x 4 x3 x5 ex arctan x 1 . . .x . . .. 1! 2! 3! 4! 3 5 Multiply these expressions and collect like terms as you would for multiplying polynomials. 1 2 1 3 1 4 . . . 1 x 2 x 6 x 24x 1 3 1 5 x 3 x 5 x . . .

2 1 3 1 4 1 5 . . . x x 2 x 6x 24 x

1 3 1 4 1 5 3 x 3 x 6 x . . .

1 5 5x . . .

2 1 3 1 4 3 5 x x 6 x 6x 40 x . . .

x 2 1 3 . . . So, e arctan x x x 6 x . b. Using the Maclaurin series for sin x and cos x in the table, you have x 3 x 5 x . . . sin x 3! 5! tan x . cos x x2 x 4 1 . . . 2! 4! Divide using long division. 1 2 x x3 x5 . . . 3 15 1 1 1 1 1 x2 x4 . . . x x3 x5 . . . 2 24 6 120 1 1 x x3 x5 . . . 2 24 1 1 x3 x5 . . . 3 30 1 1 x3 x5 . . . 3 6 2 x5 . . . 15 1 3 2 5 . . . So, tan x x 3 x 15 x . 332460_0910.qxd 11/4/04 3:12 PM Page 684

684 CHAPTER 9 Infinite Series

EXAMPLE 8 A Power Series for sin2 x

Find the power series for fx sin2 x.

Solution Consider rewriting sin2 x as follows. 1 cos 2x 1 cos 2x sin2 x 2 2 2 Now, use the series for cos x. x2 x 4 x6 x8 cos x 1 . . . 2! 4! 6! 8! 22 24 26 28 cos 2x 1 x2 x 4 x6 x8 . . . 2! 4! 6! 8! 1 1 2 23 25 27 cos 2x x2 x 4 x6 x8 . . . 2 2 2! 4! 6! 8! 1 1 1 1 2 23 25 27 sin2 x cos 2x x2 x 4 x6 x8 . . . 2 2 2 2 2! 4! 6! 8! 2 23 25 27 x2 x 4 x6 x8 . . . 2! 4! 6! 8! This series converges for < x < .

As mentioned in the preceding section, power series can be used to obtain tables of values of transcendental functions. They are also useful for estimating the values of definite integrals for which cannot be found. The next example demon- strates this use.

EXAMPLE 9 Power Series Approximation of a Definite Integral

Use a power series to approximate 1 ex2 dx 0 with an error of less than 0.01.

Solution Replacing x with x2 in the series for ex produces the following.

4 6 8 2 x x x ex 1 x2 . . . 2! 3! 4! 1 3 5 7 9 1 2 x x x x ex dx x . . . 0 3 5 2! 7 3! 9 4! 0 1 1 1 1 1 . . . 3 10 42 216 Summing the first four terms, you have 1 ex2 dx 0.74 0 1 which, by the Alternating Series Test, has an error of less than 216 0.005. 332460_0910.qxd 11/4/04 3:12 PM Page 685

SECTION 9.10 Taylor and Maclaurin Series 685

Exercises for Section 9.10 See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1–10, use the definition to find the Taylor series In Exercises 31–34, find the Maclaurin series for the function. (centered at c) for the function. (See Example 7.)

1. fx e2x, c 0 31.f x x sin x 32. hx x cos x 2. fx e3x, c 0 sin x arcsin x , x 0 , x 0 33.gx x 34. fx x 3. fx cos x, c 4 1, x 0 1, x 0 4. fx sin x, c 4 In Exercises 35 and 36, use a power series and the fact that i 2 1 to verify the formula. 5. fx ln x, c 1

x 1 6. f x e , c 1 35. gx eix e ix sin x 2i 7. fx sin 2x, c 0 36. gx 1 eix eix cos x 8. fx lnx2 1, c 0 2 9. fx sec x, c 0 (first three nonzero terms) In Exercises 37–42, find the first four nonzero terms of the 10. fx tan x, c 0 (first three nonzero terms) Maclaurin series for the function by multiplying or dividing the appropriate power series. Use the table of power series for In Exercises 11–14, prove that the Maclaurin series for the elementary functions on page 682. Use a graphing utility to function converges to the function for all x. graph the function and its corresponding polynomial approxi- mation. 11.f x cos x 12. f x e2x x x 13.fx sinh x 14. fx cosh x 37.f x e sin x 38. g x e cos x 39.hx cos x ln1 x 40. fx ex ln1 x In Exercises 15–20, use the binomial series to find the sin x ex 41.gx 42. fx Maclaurin series for the function. 1 x 1 x 1 15. fx 1 x2 In Exercises 43–46, match the polynomial with its graph. [The graphs are labeled (a), (b), (c), and (d).] Factor a common 1 16. fx factor from each polynomial and identify the function approxi- 1 x mated by the remaining Taylor polynomial. 1 17. fx (a)y (b) y 4 x2 18. fx 4 1 x 4 19. fx 1 x2 2 20. fx 1 x3 x x −4 24 −4 −2 2 4

In Exercises 21–30, find the Maclaurin series for the function. −4 −4 (Use the table of power series for elementary functions.)

2 (c)y (d) y 21. fx ex 2 22. gx e 3x 4 4 23. gx sin 3x 2 2 24. f x cos 4x x x − −4 4 4 24 − 25. fx cos x32 −2 2 − − 26. gx 2 sin x3 4 4 1 x x 27. f x 2 e e sinh x 4 3 5 x x x 28. f x e x e x 2 cosh x 43.y x2 44. y x 3! 2! 4! 29. fx cos2 x x3 30. fx sinh1 x lnx x2 1 45.y x x2 46. y x2 x3 x 4 2! 1 Hint: Integrate the series for . x2 1 332460_0910.qxd 11/4/04 3:12 PM Page 686

686 CHAPTER 9 Infinite Series

In Exercises 47 and 48, find a Maclaurin series for fx. Probability In Exercises 61 and 62, approximate the normal

x probability with an error of less than 0.0001, where the proba- 47. fx et 2 1 dt bility is given by 0 b x 1 2 x 2 3 P a < x < b e dx. 48. f x 1 t dt 2 a 0 y In Exercises 49–52, verify the sum. Then use a graphing utility 1 − 2 to approximate the sum with an error of less than 0.0001. f(x) = e x /2 2π 1 49. 1n1 ln 2 n1 n 1 50. 1n sin 1 2n 1! x n 0 ab 2n 51. e2 n! n0 61. P0 < x < 1 1 e 1 52. 1n1 62. P1 < x < 2 n1 n! e In Exercises 63–66, use a computer system to find the In Exercises 53 and 54, use the series representation of the func- fifth-degree Taylor polynomial (centered at c ) for the function. tion f to find lim fx (if it exists). x→0 Graph the function and the polynomial. Use the graph to 1 cos x determine the largest interval on which the polynomial is a 53. fx x reasonable approximation of the function. sin x 63. fx x cos 2x, c 0 54. fx x x 64. fx sin ln1 x, c 0 2 In Exercises 55–58, use a power series to approximate the value 65. gx x ln x, c 1 of the integral with an error of less than 0.0001. (In Exercises 55 3 and 56, assume that the integrand is defined as 1 when x 0. ) 66. h x x arctan x, c 1

1 sin x 55. dx Writing About Concepts 0 x 1 2 arctan x 67. State the guidelines for finding a Taylor series. 56. dx 0 x 68. If f is an even function, what must be true about the 0.3 coefficients a in the Maclaurin series 57. 1 x3 dx n 0.1 n 14 f x anx ? 58. x lnx 1 dx n 0 0 Explain your reasoning. 69. Explain how to use the series Area In Exercises 59 and 60, use a power series to approximate xn the area of the region. Use a graphing utility to verify the result. gx ex n! 2 1 n 0 59. x cos x dx 60. cos x dx to find the series for each function. Do not find the series. 0 0.5 y y (a) f x e x 3x 3 1.5 (b) f x e 4 (c) f x xex 1 2 1.0 (d) f x e2x e2x 1 70. 4 0.5 Define the binomial series. What is its radius of x convergence? π π 3π 5π x 8 4 8 8 0.5 1 1.5 332460_0910.qxd 11/4/04 3:12 PM Page 687

SECTION 9.10 Taylor and Maclaurin Series 687

71. Projectile Motion A projectile fired from the ground 76. Find the Maclaurin series for follows the trajectory given by 1 x g g kx f x ln y tan x ln 1 1 x 2 kv0 cos k v0 cos and determine its radius of convergence. Use the first four where v0 is the initial speed, is the angle of projection,g is the terms of the series to approximate ln 3. acceleration due to gravity, and k is the drag factor caused by air resistance. Using the power series representation In Exercises 77–80, evaluate the binomial coefficient using the x2 x3 x 4 formula ln1 x x . . . , 1 < x < 1 2 3 4 . . . k k k 1 k 2 k 3 k n 1 verify that the trajectory can be rewritten as n n! gx2 kgx3 k2 gx4 where k is a real number, n is a positive integer, and y tan x . . . . v 2 2 v 3 3 v 4 4 2 0 cos 3 0 cos 4 0 cos k 1. 72. Projectile Motion Use the result of Exercise 71 to determine 0 the series for the path of a projectile launched from ground 5 2 level at an angle of 60 , with an initial speed of v0 64 77. 78. 1 3 2 feet per second and a drag factor of k 16. 0.5 13 73. Investigation Consider the function f defined by 79. 80. 4 5 e1x2, x 0 fx k 0, x 0. 81. Write the power series for 1 x in terms of binomial coefficients. (a) Sketch a graph of the function. 82. Prove that e is irrational. Hint: Assume that e pq is (b) Use the alternative form of the definition of the derivative rational (p and q are integers) and consider (Section 2.1) and L’Hôpital’s Rule to show that f0 0. [By continuing this process, it can be shown that f n0 1 1 e 1 1 . . . . . . . 0 for n > 1. ] 2! n! f (c) Using the result in part (b), find the Maclaurin series for . 83. Show that the Maclaurin series of the function Does the series converge to f ? x 74. Investigation gx 1 x x2 (a) Find the power series centered at 0 for the function is 2 ln x 1 f x 2 . x n Fn x n1 (b) Use a graphing utility to graph f and the eighth-degree where F is the n th Fibonacci number with F F 1 and Taylor polynomial P8 x for f. n 1 2 F F F , for n ≥ 3. (c) Complete the table, where n n2 n1 Hint: Write x 2 x ln t 1 F x 2 dt and G x P8 t dt. x 0 t 0 a a x a x2 . . . 1 x x2 0 1 2

x 0.25 0.50 0.75 1.00 1.50 2.00 and multiply each side of this equation by 1 x x2. Fx Gx Putnam Exam Challenge

(d) Describe the relationship between the graphs of f and P8 84. Assume that fx ≤ 1 and fx ≤ 1 for all x on an interval and the results given in the table in part (c). of length at least 2. Show that fx ≤ 2 on the interval. xn 75. Prove that lim 0 for any real x. This problem was composed by the Committee on the Putnam Prize Competition. n→ n! ©The Mathematical Association of America. All rights reserved.