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

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CHAPTER 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 CHAPTER 8 Infinite Series Section 8.1 Sequences Solutions to Odd-Numbered Exercises 1. a 5 5 2 5 5 32 a 4 5 2 4 5 16 a 3 5 2 3 5 8 a 2 5 2 2 5 4 a 1 5 2 1 5 2 a n 5 2 n 3. a 5 5 1 2 1 2 2 5 52 1 32 a 4 5 1 2 1 2 2 4 5 1 16 a 3 5 1 2 1 2 2 3 1 8 a 2 5 1 2 1 2 2 2 5 1 4 a 1 5 1 2 1 2 2 1 1 2 a n 5 1 2 1 2 2 n 5. a 5 5 sin 5 p 2 5 1 a 4 5 sin 2 5 0 a 3 5 sin 3 2 1 a 2 5 sin 5 0 a 1 5 sin 2 5 1 a n 5 sin n 2 7. a 5 5 s 2 1 d 15 5 2 1 25 a 4 5 s 2 1 d 10 4 2 5 1 16 a 3 5 s 2 1 d 6 3 2 5 1 9 a 2 5 s 2 1 d 3 2 2 1 4 a 1 5 s 2 1 d 1 1 2 1 a n 5 s 2 1 d n s n 1 1 d y 2 n 2 9. a 5 5 5 2 1 5 1 1 25 5 121 25 a 4 5 5 2 1 4 1 1 16 5 77 16 a 3 5 5 2 1 3 1 1 9 5 43 9 a 2 5 5 2 1 2 1 1 4 5 19 4 a 1 5 5 2 1 1 1 5 5 a n 5 5 2 1 n 1 1 n 2 11. a 5 5 3 5 5! 5 243 120 a 4 5 3 4 4! 5 81 24 a 3 5 3 3 3! 5 27 6 a 2 5 3 2 2! 5 9 2 a 1 5 3 1! 5 3 a n 5 3 n n ! 13. 5 2 s 10 2 1 d 5 18 a 5 5 2 s a 4 2 1 d 5 2 s 6 2 1 d 5 10 a 4 5 2 s a 3 2 1 d 5 2 s 4 2 1 d 5 6 a 3 5 2 s a 2 2 1 d 5 2 s 3 2 1 d 5 4 a 2 5 2 s a 1 2 1 d a 1 5 3, a k 1 1 5 2 s a k 2 1 d 15. a 5 5 1 2 a 4 5 1 2 s 4 d 5 2 a 4 5 1 2 a 3 5 1 2 s 8 d 5 4 a 3 5 1 2 a 2 5 1 2 s 16 d 5 8 a 2 5 1 2 a 1 5 1 2 s 32 d 5 16 a 1 5 32, a k 1 1 5 1 2 a k
17. Because and the sequence matches graph (d). a 2 5 8 y s 2 1 1 d 5 8 3 , a 1 5 8 y s 1 1 1 d 5 4 19. This sequence decreases and Matches (c). a 2 5 4 s 0.5 d 5 2. a 1 5 4, 21. a n 5 2 3 n , n 5 1, . . . , 10 1 1 12 8 23. a n 5 16 s 2 0.5 d n 2 1 , n 5 1, . . . , 10 12 1 10 18 25. a n 5 2 n n 1 1 , n 5 1, 2, . . . , 10 12 1 1 3 27. Add 3 to preceeding term. a 6 5 3 s 6 d 2 1 5 17 a 5 5 3 s 5 d 2 1 5 14 a n 5 3 n 2 1 29. Multiply the preceeding term by 2 1 2 . a 6 5 3 s 2 2 d 5 52 3 32 a n 5 3 s 2 2 d 4 5 3 16 a n 5 3 s 2 2 d n 2 1 31. 5 s 9 ds 10 d 5 90 10! 8! 5 8! s 9 ds 10 d 8! 33. 5 n 1 1 s n 1 1 d ! n ! 5 n ! s n 1 1 d n ! 35. 5 1 2 n s 2 n 1 1 d s 2 n 2 1 d ! s 2 n 1 1 d ! 5 s 2 n 2 1 d ! s 2 n 2 1 d ! s 2 n ds 2 n 1 1 d 37. lim n ` 5 n 2 n 2 1 2 5 5 39. 5 2 1 5 2 lim n ` 2 n ! n 2 1 1 5 lim n ` 2 ! 1 1 s 1 y n 2 d 41. lim n ` sin 1 1 n 2 5 0 43. The graph seems to indicate that the sequence converges to 1. Analytically, lim n ` a n 5 lim n ` n 1 1 n 5 lim x ` x 1 1 x 5 lim x ` 1 5 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 H a n J 5 H 0, 2 1, 0, 1, 0, 2 1, . . . J . 12 1 2 2 47. does not exist (oscillates between and 1), diverges. 2 1 lim n ` s 2 1 d n 1 n n 1 1 2 49. converges lim n ` 3 n 2 2 n 1 4 2 n 2 1 1 5 3 2 , 51. converges lim n ` 1 1 s 2 1 d n n 5 0, 53.

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## This note was uploaded on 05/18/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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

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