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ODD08 - CHAPTER Infinite Series Section 8.1 Section 8.2...

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C H A P T E R 8 Infinite Series Section 8.1 Sequences . . . . . . . . . . . . . . . . . . . . . 121 Section 8.2 Series and Convergence . . . . . . . . . . . . . . 126 Section 8.3 The Integral Test and p -Series . . . . . . . . . . 131 Section 8.4 Comparisons of Series . . . . . . . . . . . . . . 135 Section 8.5 Alternating Series . . . . . . . . . . . . . . . . . 138 Section 8.6 The Ratio and Root Tests . . . . . . . . . . . . . 142 Section 8.7 Taylor Polynomials and Approximations . . . . . 147 Section 8.8 Power Series . . . . . . . . . . . . . . . . . . . . 152 Section 8.9 Representation of Functions by Power Series . . 157 Section 8.10 Taylor and Maclaurin Series . . . . . . . . . . . 160 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 167 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . 172
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121 C H A P T E R 8 Infinite Series Section 8.1 Sequences Solutions to Odd-Numbered Exercises 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 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. a 5 3 5 5! 243 120 a 4 3 4 4! 81 24 a 3 3 3 3! 27 6 a 2 3 2 2! 9 2 a 1 3 1! 3 a n 3 n n ! 13. 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 15. 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
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17. Because and the sequence matches graph (d). a 2 8 2 1 8 3 , a 1 8 1 1 4 19. This sequence decreases and Matches (c). a 2 4 0.5 2. a 1 4, 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. a n 2 n n 1 , n 1, 2, . . . , 10 12 1 1 3 27. Add 3 to preceeding term. a 6 3 6 1 17 a 5 3 5 1 14 a n 3 n 1 29. Multiply the preceeding term by 1 2 . a 6 3 2 5 3 32 a n 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 1 ! n ! n ! n 1 n ! 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 39. 2 1 2 lim n 2 n n 2 1 lim n 2 1 1 n 2 41. lim n sin 1 n 0 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 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. converges lim n 1 1 n n 0, 53. 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 122 Chapter 8 Infinite Series
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55. converges lim n 3 4 n 0, 57. diverges lim n n 1 ! n ! lim n n 1 , 59. 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 61. converges p > 0, n 2 lim n n p e n 0, 63. 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 65. converges lim n sin n n lim n sin n 1 n 0, 67. a n 3 n 2 69. a n n 2 2 71. a n n 1 n 2 73. a n 1 n 1 2 n 2 75. a n 1 1 n n 1 n 77. a n n n 1 n 2 79. a n 1 n 1 1 3 5 . . .
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