# EVEN08 - 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 . . . . . . . . . . . . . . . . . . . . . 369 Section 8.2 Series and Convergence . . . . . . . . . . . . . . 373 Section 8.3 The Integral Test and p -Series . . . . . . . . . . 378 Section 8.4 Comparisons of Series . . . . . . . . . . . . . . 381 Section 8.5 Alternating Series . . . . . . . . . . . . . . . . . 385 Section 8.6 The Ratio and Root Tests . . . . . . . . . . . . . 389 Section 8.7 Taylor Polynomials and Approximations . . . . . 393 Section 8.8 Power Series . . . . . . . . . . . . . . . . . . . . 398 Section 8.9 Representation of Functions by Power Series . . 403 Section 8.10 Taylor and Maclaurin Series . . . . . . . . . . . 408 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 414 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . 421
369 C H A P T E R 8 Infinite Series Section 8.1 Sequences Solutions to Even-Numbered Exercises 2. a 5 10 8 5 4 a 4 8 7 a 3 6 6 1 a 2 4 5 a 1 2 4 1 2 a n 2 n n 3 4. a 5 32 243 a 4 16 81 a 3 8 27 a 2 4 9 a 1 2 3 a n 2 3 n 6. a 5 cos 5 2 0 a 4 cos 2 1 a 3 cos 3 2 0 a 2 cos 1 a 1 cos 2 0 a n cos n 2 8. a 5 2 5 a 4 2 4 1 2 a 3 2 3 a 2 2 2 1 a 1 2 1 2 a n 1 n 1 2 n 10. a 5 10 2 5 6 25 266 25 a 4 10 1 2 3 8 87 8 a 3 10 2 3 2 3 34 3 a 2 10 1 3 2 25 2 a 1 10 2 6 18 a n 10 2 n 6 n 2 12. a 5 3 5 15 a 4 3 4 12 a 3 3 3 9 a 2 3 2 6 a 1 3 1 3 a n 3 n ! n 1 ! 3 n 14. a 5 4 1 2 a 4 30 a 4 3 1 2 a 3 12 a 3 2 1 2 a 2 6 a 2 1 1 2 a 1 4 a 1 4, a k 1 k 1 2 a k 16. a 5 1 3 a 4 2 1 3 768 2 196,608 a 4 1 3 a 3 2 1 3 48 2 768 a 3 1 3 a 2 2 1 3 12 2 48 a 2 1 3 a 1 2 1 3 6 2 12 a 1 6, a k 1 1 3 a k 2
18. Because the sequence tends to 8 as n tends to infinity, it matches (a). 20. This sequence increases for a few terms, then decreases Matches (b). a 2 16 2 8. 22. n 1, . . . , 10 a n 2 4 n , 1 12 3 4 24. a n 8 0.75 n 1 , n 1, 2, . . . , 10 1 12 1 10 26. a n 3 n 2 n 2 1 , n 1, . . . , 10 1 12 1 4 28. a 6 6 6 2 6 a 5 5 6 2 11 2 a n n 6 2 30. a 6 2 80 160 a 5 2 40 80 a 1 5 a n 1 2 a n , 32. 24 25 600 25! 23! 23! 24 25 23! 34. n 1 n 2 n 2 ! n ! n ! n 1 n 2 n ! 36. 2 n 1 2 n 2 2 n 2 ! 2 n ! 2 n ! 2 n 1 2 n 2 2 n ! 38. lim n 5 1 n 2 5 0 5 40. 5 1 5 lim n 5 n n 2 4 lim n 5 1 4 n 2 42. lim n cos 2 n 1 44. The graph seems to indicate that the sequence converges to 0. Analytically, lim n a n lim n 1 n 3 2 lim x 1 x 3 2 0. 1 12 1 2 46. The graph seems to indicate that the sequence converges to 3. Analytically, lim n a n lim n 3 1 2 n 3 0 3. 1 12 1 4 48. does not exist, (alternates between 0 and 2), diverges. lim n 1 1 n 50. converges lim n 3 n 3 n 1 1, 52. converges lim n 1 1 n n 2 0, 54. converges (L’Hôpital’s Rule) lim n 1 2 n 0, lim n ln n n lim n 1 2 ln n n 56. converges lim n 0.5 n 0, 58. converges lim n n 2 ! n ! lim n 1 n n 1 0, 370 Chapter 8 Infinite Series
60. converges lim n n 2 2 n 1 n 2 2 n 1 lim n 2 n 2 4 n 2 1 1 2 , 62. Let (L’Hôpital’s Rule) or, Therefore lim n n sin 1 n 1. lim x sin 1 x 1 x lim y 0 sin y y 1. lim x x sin 1 x lim x sin 1 x 1 x lim x 1 x 2 cos 1 x 1 x 2 lim x cos 1 x cos 0 1 f x x sin 1 x . a n n sin 1 n 64. converges lim n 2 1 n 2 0 1, 66. converges lim n cos n n 2 0, 68. a n 4 n 1 70. a n 1 n 1 n 2 72. a n n 2 3 n 1 74. a n 1 n 3 n 2 2 n 1 76.