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C0900TextPages_page_1 Charles & Josette Lenars/Corbis 593 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated,…

C0900TextPages_page_10 602 CHAPTER 9 Infinite Series com for works In Exercises 1-10, write the first five terms of the sequence. a] In Exercises 21-24, use a…

C0900TextPages_page_100 692 CHAPTER 9 Infinite Series 8. Find f"?(0) if f(x) =e*. (Hint: Do not calculate 12 derivatives.) 9. The graph of the function L x=0 FQ) =…

C0900TextPages_page_11 eed" 4 (-1 sao, = been st o,-1t EU" n n In(n*) In Jn 55. == 56. =—_ a = an a, n 3" fe 57.4, =, 58. a, = (0.5) ! —2)! 59.4 = (n + 1)! 60.…

C0900TextPages_page_12 604 CHAPTER 9 Infinite Series Writing About Concepts (continued) In Exercises 105-108, give an example of a sequence satisfying the…

C0900TextPages_page_13 122. 123. 124. 125. 126. 127. 128. Conjecture Let x) = 1 and consider the sequence x, given by the formula 1 1 +—, n=1,2,.... X, , Xn a in…

C0900TextPages_page_14 606 CHAPTER 9 INFINITE SERIES. The study of infinite series was considered a novelty in the fourteenth century. Logician Richard Suiseth,…

C0900TextPages_page_15 TECHNOLOGY Figure 9.5 shows the first 15 partial sums of the infinite series in Example 1(a). Notice how the values appear to approach the…

C0900TextPages_page_16 608 CHAPTER 9 Infinite Series EXAMPLE 2 Writing a Series in Telescoping Form aS 2 Find the sum of the series > dent: Solution Using partial…

C0900TextPages_page_17 SECTION 9.2 Series and Convergence 609 TECHNOLOGY | Try using a EXAMPLE 3 Convergent and Divergent Geometric Series graphing utility or…

C0900TextPages_page_18 610 CHAPTER 9 Infinite Series ‘STUDY TIP As you study this chapter, The following properties are direct consequences of the corresponding…

C0900TextPages_page_19 SECTION 9.2 Series and Convergence oll EXAMPLE 5 Using the nth-Term Test for Divergence oo a. For the series > 2”, you have n=0 lim 2” =…

C0900TextPages_page_2 594 CHAPTER 9 Infinite Series Finding Patterns Describe a pattern for each of the following sequences. Then use your description to write a…

C0900TextPages_page_20 Infinite Series es for Section 9.2 612 CHAPTER 9 In Exercises 1-6, find the first five terms of the sequence of partial sums. P,l,lydty..…

C0900TextPages_page_21 & i ed 8 37. Gt Dine d) 38. Gn F Qn +3) «2 /1\' /4\" » 303) ». S63) ea Vv 2 ve 41. S( 5) 42. S2( ;) 43. 1 + 0.1 + 0.01 + 0.001 ++ - =…

C0900TextPages_page_22 614 CHAPTER 9 Infinite Series a Writing In Exercises 93 and 94, use a graphing utility to deter- 103. Probability A fair coin is tossed…

C0900TextPages_page_23 108. Sphereflake A sphereflake shown below is a computer- 115. generated fractal that was created by Eric Haines. The radius of the large…

C0900TextPages_page_24 616 CHAPTER 9 Infinite Series a 2 | 128. If S) a, converges where a, is nonzero, show that 5) — n=! n=1"n diverges. 129. The Fibonacci…

C0900TextPages_page_25 SECTION 9.3 The Integral Test and p-Series 617 The Integral Test and p-Series ¢ Use the Integral Test to determine whether an infinite…

C0900TextPages_page_26 618 CHAPTER 9 Infinite Series EXAMPLE | Using the Integral Test n Apply the Integral Test to the series > PEL n=l Solution The function…

C0900TextPages_page_27 HARMONIC SERIES Pythagoras and his students paid close attention to the development of music as an abstract science. This led to the…

C0900TextPages_page_28 620 CHAPTER 9 Infinite Series EXAMPLE 4 Testing a Series for Convergence Determine whether the following series converges or diverges. 1 »…

C0900TextPages_page_29 In Exercises 21-24, explain why the Integral Test does not apply to the series. & (-1)" eA 2. > 22. 5) e" cos n mm nt S 2+ sinn & /sinn\? a…

C0900TextPages_page_3 1234560 M Forn > M, the terms of the sequence all lie within € units of L. Figure 9.1 NOTE There are different ways in which a sequence can…

C0900TextPages_page_30 622 CHAPTER 9 Infinite Series 60. Show that the result of Exercise 59 can be written as Writing About Concepts (continued) . N <2 N 48. In…

C0900TextPages_page_31 75. Euler’s Constant Let | 1 1 S,= Nosltstee te. a 2 n (a) Show that In(n + 1) < S, < 1+ Inn. (b) Show that the sequence {a,,} = {5,, —…

C0900TextPages_page_32 624 CHAPTER 9 Infinite Series ‘ons of Series Compa ¢ Use the Direct Comparison Test to determine whether a series converges or diverges. ©…

C0900TextPages_page_33 SECTION 9.4 Comparisons of Series 625 EXAMPLE | _ Using the Direct Comparison Test Determine the convergence or divergence of 2 1 224…

C0900TextPages_page_34 626 CHAPTER 9 Infinite Series NOTE As with the Direct Comparison Test, the Limit Comparison Test could be modified to require only that a,,…

C0900TextPages_page_35 SECTION 9.4 Comparisons of Series 627 The Limit Comparison Test works well for comparing a “messy” algebraic series with a p-series. In…

C0900TextPages_page_36 628 CHAPTER 9 Infinite Series Section 9.4 es fo com for ork 1. Graphical Analysis The figures show the graphs of the first 10 terms, and…

C0900TextPages_page_37 SECTION 9.4 Comparisons of Series 629 37. Use the Limit Comparison Test with the harmonic series to sae . show that the series = a, (where…

C0900TextPages_page_38 630 CHAPTER 9 Infinite Series 60. If0 b,, diverges, then Ss d,, diverges. 1 1 61. Prove that if the nonnegative series oo s…

C0900TextPages_page_39 SECTION 9.5 Alternating Series 631 Alternating Series ¢ Use the Alternating Series Test to determine whether an infinite series converges.…

C0900TextPages_page_4 596 CHAPTER 9 Infinite Series The following properties of limits of sequences parallel those given for limits of functions of a real…

C0900TextPages_page_40 632 CHAPTER 9 Infinite Series EXAMPLE | Using the Alternating Series Test ed 1 NOTE The series in Example 1 Determine the convergence or…

C0900TextPages_page_41 SECTION 9.5 Alternating Series 633 Alternating Series Remainder For a convergent alternating series, the partial sum Sy can be a useful…

C0900TextPages_page_42 634 CHAPTER 9 Infinite Series Absolute and Conditional Convergence Occasionally, a series may have both positive and negative terms and not…

C0900TextPages_page_43 SECTION 9.5 Alternating Series 635 EXAMPLE 5 Absolute and Conditional Convergence Determine whether each of the series is convergent or…

C0900TextPages_page_44 636 CHAPTER 9 Infinite Series EXAMPLE7 Rearrangement of a Series FOR FURTHER INFORMATION Georg The alternating harmonic series converges to…

C0900TextPages_page_45 In Exercises 11-32, determine the convergence or divergence of the series. @ cin! @ (-1)"*In iu. > 2. ¥ ——* ma on i 2n- 1 & (ciyett @ (=)…

C0900TextPages_page_46 638 CHAPTER 9 Infinite Series True or False? In Exercises 67-70, determine whether the statement is true or false. If it is false, explain…

C0900TextPages_page_47 SECTION 9.6 The Ratio and Root Tests 639 he Ratio and Root Tests Writing a Series One of the following conditions guarantees that a series…

C0900TextPages_page_48 640 CHAPTER 9 Infinite Series Although the Ratio Test is not a cure for all ills related to tests for convergence, it is particularly…

C0900TextPages_page_49 SECTION 9.6 The Ratio and Root Tests 641 Goats EXAMPLE 3 A Failure of the Ratio Test vn n+ oo Determine the convergence or divergence of >…

C0900TextPages_page_5 Forn = 4, (—1)"/n! is squeezed between —1/2"and 1/2". Figure 9.2 NOTE Example 5 suggests something about the rate at which n! increases as…

C0900TextPages_page_50 642 CHAPTER 9 Infinite Series NOTE The Root Test is always incon- clusive for any p-series. FOR FURTHER INFORMATION For more information on…

C0900TextPages_page_51 SECTION 9.6 The Ratio and Root Tests 643 Strategies for Testing Series You have now studied 10 tests for determining the convergence or…

C0900TextPages_page_52 644 CHAPTER 9 Infinite Series Summary of Tests for Series Test Series Condition(s) Condition(s) Comment of Convergence of Divergence…

C0900TextPages_page_53 SECTION 9.6 The Ratio and Root Tests 645 es for Sec C ‘out solutions to odd-numbered exe! In Exercises 1-4, verify the formula. Pe…

C0900TextPages_page_54 646 CHAPTER 9 Infinite Series In Exercises 37-50, use the Root Test to determine the conver- gence or divergence of the series. 37. 39. 4.…

C0900TextPages_page_55 In Exercises 87-92, find the values of x for which the series converges. v.33) ® SS) oS Cure + ay 90. Sax — 1)" n=0 91. x m(3)! m Sn…

C0900TextPages_page_56 648 CHAPTER 9 Infinite Series Taylor Polynomials and Approximations P(c)=f(c) P(O=f(O Near (c, f(c)), the graph of P can be used to…

C0900TextPages_page_57 SECTION 9.7 Taylor Polynomials and Approximations 649 In Figure 9.12 you can see that, at points near (0, 1), the graph of P\(x) =1 +x…

C0900TextPages_page_58 650 CHAPTER 9 Brook TavLor (1685-1731) Although Taylor was not the first to seek polynomial approximations of transcendental functions, his…

C0900TextPages_page_59 SECTION 9.7 Taylor Polynomials and Approximations 651 EXAMPLE 4_ Finding Taylor Polynomials for In x Find the Taylor polynomials P,, P,,…

C0900TextPages_page_6 598 CHAPTER 9 Infinite Series In Example 5, the sequence {c,} has both positive and negative terms. For this sequence, it happens that the…

C0900TextPages_page_60 652 CHAPTER 9 F(x) = cos x Near (0, 1), the graph of P, can be used to approximate the graph of f(x) = cos x. Figure 9.15 Infinite Series…

C0900TextPages_page_61 SECTION 9.7 Taylor Polynomials and Approximations 653 Taylor polynomials and Maclaurin polynomials can be used to approximate the value of…

C0900TextPages_page_62 654 CHAPTER 9 Infinite Series Remainder of a Taylor Polynomial An approximation technique is of little value without some idea of its…

C0900TextPages_page_63 NOTE Try using a calculator to verify the results obtained in Examples 8 and 9. For Example 8, you obtain sin(0.1) ~ 0.0998334. For Example…

C0900TextPages_page_64 656 CHAPTER 9 Infinite Series es for Section 9.7 0 odd-numbered exercises In Exercises 1-4, match the Taylor polynomial approximation 7 of…

C0900TextPages_page_65 SECTION 9.7 Taylor Polynomials and Approximations 657 ad In Exercises 31 and 32, use a computer algebra system to find In Exercises 37-40,…

C0900TextPages_page_66 658 CHAPTER 9 Infinite Series 84. f(x) = cos(a x’), approximate f(0.6). 85. f(x) = e-™, approximate f(1.3) 56. f(x) = e7*, approximate…

C0900TextPages_page_67 SECTION 9.8 Power Series 659 ¢ Understand the definition of a power series. ¢ Find the radius and interval of convergence of a power…

C0900TextPages_page_68 660 CHAPTER 9 Infinite Series A single point + x c An interval EEE caeeeesaeeeee EEE x c on cnn R R The real line $$ e—o x — c The domain…

C0900TextPages_page_69 SECTION 9.8 Power Series 661 STUDY TIP To determine the radius of © EXAMPLE 2 Finding the Radius of Convergence convergence of a power…

C0900TextPages_page_7 SECTION 9.1 Sequences 599 Without a specific rule for generating the terms of a sequence or some knowledge of the context in which the…

C0900TextPages_page_70 662 CHAPTER 9 Infinite Series Endpoint Convergence Note that for a power series whose radius of convergence is a finite number R, Theorem…

C0900TextPages_page_71 SECTION 9.8 Power Series 663 EXAMPLE 6 Finding the Interval of Convergence —1)"(x + 1)" Find the interval of convergence of > Cures n=0…

C0900TextPages_page_72 664 CHAPTER 9 Infinite Series Differentiation and Integration of Power Series Power series representation of functions has played an…

C0900TextPages_page_73 SECTION 9.8 Power Series 665 EXAMPLE 8 Intervals of Convergence for f(x), f(x), and f f(x) dx Consider the function given by eo yt x? x…

C0900TextPages_page_74 666 CHAPTER 9 Infinite Series es for Section 9.8 In Exercises 1-4, state where the power series is centered. Ss <3 (-1)"1 +3 1. Pes 2. >»…

C0900TextPages_page_75 49. g(1) 50. g(2) 51. (3.1) 52. g(—2) Writing In Exercises 53-56, match the graph of the first 10 terms of the sequence of partial sums of…

C0900TextPages_page_76 668 CHAPTER 9 Infinite Series 72. Bessel Function The Bessel function of order 1 is oe (= 1k x2 J{(x) = Seo an 1) , PIM + D! (a) Show that…

C0900TextPages_page_77 SECTION 9.9 Representation of Functions by Power Series 669 Representation of Functions by Power Series e Find a geometric power series…

C0900TextPages_page_78 670 CHAPTER 9 Infinite Series EXAMPLE | Finding a Geometric Power Series Centered at 0 Find a power series for f(x) = ya? centered at 0.…

C0900TextPages_page_79 SECTION 9.9 Representation of Functions by Power Series 671 Operations with Power Series The versatility of geometric power series will be…

C0900TextPages_page_8 600 CHAPTER 9 Infinite Series Joe aN t oy I 34 toy / \ol at ow be! a a, I> {a,}= (3+ CD") _ $+ + + > n by, x at 3 2n b,} = {22 d= (Thad ot…

C0900TextPages_page_80 672 CHAPTER 9 Infinite Series EXAMPLE 4 Finding a Power Series by Integration Find a power series for f(x) = In x, centered at 1. Solution…

C0900TextPages_page_81 SECTION 9.9 Representation of Functions by Power Series 673 Gout D EXAMPLE 5 Finding a Power Series by Integration 5 5 3 3 3 ou g 5 se) 2 i…

C0900TextPages_page_82 674 CHAPTER 9 Infinite Series es for Section 9.9 s to odd-numbered exercises In Exercises 1-4, find a geometric power series for the…

C0900TextPages_page_83 In Exercises 35-38, use the series for f(x) = arctan x to approx- imate the value, using Ry < 0.001. 1 35. arctan 4 37. ie arctan x? ik 0 x…

C0900TextPages_page_84 676 CHAPTER 9 n 9.10 Bettmann/Corbis COLIN MACLAURIN (1698-1746) The development of power series to represent functions is credited to the…

C0900TextPages_page_85 SECTION 9.10 Taylor and Maclaurin Series 677 Definitions of Taylor and Maclaurin Series Tf a function f has derivatives of all orders at x…

C0900TextPages_page_86 678 CHAPTER 9 Infinite Series -1 x< =5 fe =4sinx, — |x| < 5 1, x> £ Figure 9.23 Notice that in Example | you cannot conclude that the power…

C0900TextPages_page_87 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…

C0900TextPages_page_88 680 CHAPTER 9 Infinite Series The guidelines for finding a Taylor series for f(x) at c are summarized below. Guidelines for Finding a…

C0900TextPages_page_89 SECTION 9.10 Taylor and Maclaurin Series 681 Binomial Series Before presenting the basic list for elementary functions, you wll develop one…

C0900TextPages_page_9 NOTE All three sequences shown in Figure 9.3 are bounded. To see this, consider the following. 2

C0900TextPages_page_90 682 CHAPTER 9 Infinite Series Deriving Taylor Series from a Basic List The following list provides the power series for several elementary…

C0900TextPages_page_91 SECTION 9.10 Taylor and Maclaurin Series 683 Power series can be multiplied and divided like polynomials. After finding the first few terms…

C0900TextPages_page_92 684 CHAPTER 9 Infinite Series EXAMPLE 8 A Power Series for sin? x Find the power series for f(x) = sin? x. Solution Consider rewriting sin?…

C0900TextPages_page_93 es for Sec SECTION 9.10 Taylor and Maclaurin Series 685 ‘Out solutions to odd-numbered exe In Exercises 1-10, use the definition to find…

C0900TextPages_page_94 686 CHAPTER 9 Infinite Series In Exercises 47 and 48, find a Maclaurin series for f(x). , 47. fx) = I (e" = 1dr lo 48. n= | V1 + dt 0 In…

C0900TextPages_page_95 71. Projectile Motion A projectile fired from the ground follows the trajectory given by - g g kx ) >= (tane— — nl - y (1an 9 kyo cos ) *…

C0900TextPages_page_96 688 CHAPTER 9 Infinite Series com for works Review Exercises for Chapter In Exercises 1 and 2, write an expression for the nth term of the…

C0900TextPages_page_97 REVIEW EXERCISES 689 In Exercises 31 and 32, (a) write the repeating decimal as a & Numerical, Graphical, and Analytic Analysis In…

C0900TextPages_page_98 690 CHAPTER 9 Infinite Series In Exercises 65-70, find the interval of convergence of the In Exercises 87-92, find the sum of the…

C0900TextPages_page_99 | Ps. | Problem Solvi 1. The Cantor set (Georg Cantor, 1845-1918) is a subset of the unit interval [0, 1]. To construct the Cantor set,…