# Chapter9 - CHAPTER Infinite Series 9 Section 9.1 Sequences...

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C H A P T E R 9 Infinite Series Section 9.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 437 Section 9.2 Series and Convergence . . . . . . . . . . . . . . . . . . . 444 Section 9.3 The Integral Test and p -Series . . . . . . . . . . . . . . . . 451 Section 9.4 Comparisons of Series . . . . . . . . . . . . . . . . . . . . 457 Section 9.5 Alternating Series . . . . . . . . . . . . . . . . . . . . . . 460 Section 9.6 The Ratio and Root Tests . . . . . . . . . . . . . . . . . . 465 Section 9.7 Taylor Polynomials and Approximations . . . . . . . . . . 472 Section 9.8 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . 477 Section 9.9 Representation of Functions by Power Series . . . . . . . 483 Section 9.10 Taylor and Maclaurin Series . . . . . . . . . . . . . . . . 487 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

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C H A P T E R 9 Infinite Series Section 9.1 Sequences 437 9. a 5 5 1 5 1 25 121 25 a 4 5 1 4 1 16 77 16 a 3 5 1 3 1 9 43 9 a 2 5 1 2 1 4 19 4 a 1 5 1 1 5 a n 5 1 n 1 n 2 11. 2 10 1 18 a 5 2 a 4 1 2 6 1 10 a 4 2 a 3 1 2 4 1 6 a 3 2 a 2 1 2 3 1 4 a 2 2 a 1 1 a 1 3, a k 1 2 a k 1 13. a 5 1 2 a 4 1 2 4 2 a 4 1 2 a 3 1 2 8 4 a 3 1 2 a 2 1 2 16 8 a 2 1 2 a 1 1 2 32 16 a 1 32, a k 1 1 2 a k 15. decreases to 0; matches (f). a n 8 n 1 , a 1 4, a 2 8 3 , 17. decreases to 0; matches (e). a n 4 0.5 n 1 , a 1 4, a 2 2, 19. etc.; matches (d). a 3 1, a n 1 n , a 1 1, a 2 1, 1. a 5 2 5 32 a 4 2 4 16 a 3 2 3 8 a 2 2 2 4 a 1 2 1 2 a n 2 n 3. a 5 1 2 5 1 32 a 4 1 2 4 1 16 a 3 1 2 3 1 8 a 2 1 2 2 1 4 a 1 1 2 1 1 2 a n 1 2 n 5. a 5 sin 5 2 1 a 4 sin 2 0 a 3 sin 3 2 1 a 2 sin 0 a 1 sin 2 1 a n sin n 2 7. a 5 1 15 5 2 1 25 a 4 1 10 4 2 1 16 a 3 1 6 3 2 1 9 a 2 1 3 2 2 1 4 a 1 1 1 1 2 1 a n 1 n n 1 2 n 2 21. a n 2 3 n , n 1, . . . , 10 1 1 12 8 23. a n 16 0.5 n 1 , n 1, . . . , 10 12 1 10 18 25. Add 3 to preceding term. a 6 3 6 1 17 a 5 3 5 1 14 a n 3 n 1 27. a 6 2 80 160 a 5 2 40 80 a 1 5 a n 1 2 a n ,
438 Chapter 9 Infinite Series 43. The graph seems to indicate that the sequence converges to 1. Analytically, lim n a n lim n n 1 n lim x x 1 x lim x 1 1. 1 1 12 3 45. The graph seems to indicate that the sequence diverges. Analytically, the sequence is Hence, does not exist. lim n a n a n 0, 1, 0, 1, 0, 1, . . . . 12 1 2 2 29. Multiply the preceding term by 1 2 . a 6 3 2 5 3 32 a 5 3 2 4 3 16 a n 3 2 n 1 31. 9 10 90 10! 8! 8! 9 10 8! 33. n 1 ! n ! n ! n 1 n ! n 1 39. lim n 2 n n 2 1 lim n 2 1 1 n 2 2 1 2 41. lim n sin 1 n 0 47. does not exist (oscillates between and 1), diverges. 1 lim n 1 n n n 1 49. converges lim n 3 n 2 n 4 2 n 2 1 3 2 , 51. Thus, converges. lim n a n 0, 1 2 n 3 2 n 5 2 n . . . 2 n 1 2 n < 1 2 n a n 1 3 5 . . . 2 n 1 2 n n 53. converges lim n 1 1 n n 0, 35. 1 2 n 2 n 1 2 n 1 ! 2 n 1 ! 2 n 1 ! 2 n 1 ! 2 n 2 n 1 37. lim n 5 n 2 n 2 2 5 55. converges (L’Hôpital’s Rule) lim n 3 2 1 n 0, lim n ln n 3 2 n lim n 3 2 ln n n 57. converges lim n 3 4 n 0, 59. , diverges lim n n 1 ! n ! lim n n 1 61. converges lim n 1 2 n n 2 n 0, lim n n 1 n n n 1 lim n n 1 2 n 2 n n 1

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Section 9.1 Sequences 439 63. converges p > 0, n 2 lim n n p e n 0, 65. where converges u k n , lim n 1 k n n lim u 0 1 u 1 u k e k a n 1 k n n 67. converges (because is bounded) sin n lim n sin n n lim n sin n 1 n 0, 69. a n 3 n 2 71. a n n 2 2 73. a n n 1 n 2 75. a n 1 1 n n 1 n 77. a n n n 1 n 2 79.
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