EVNREV08 - 414 66 a2n Chapter 8 Infinite Series 68 Answers...

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68. Answers will vary. 70. 3 x 32 n 2 2 n x n n 64 n 16 n 2 3 x 32 n 2 x n n 32 n 16 n 2 3 x 32 2 2 x 2 2 64 2 2 3 x 3 3 64 3 16 2 4 x 4 4 64 4 16 2 . . . y 3 x 32 x 2 2 64 2 1 2 2 1 16 32 x 3 3 64 3 1 2 3 1 16 2 32 x 4 4 64 4 1 2 4 . . . 60 , v 0 64, k 1 16 , g 32 72. (a) From Exercise 8, you obtain (c) (d) The curves are nearly identical for Hence, the integrals nearly agree on that interval. 0 < x < 1. G x x 0 P 8 t dt F x x 0 ln t 2 1 t 2 dt P 8 1 x 2 2 x 4 3 x 6 4 x 8 5 P 1 x 2 n 0 1 n x 2 n 2 n 1 n 0 1 n x 2 n n 1 f x ln x 2 1 x 2 . x 0.25 0.50 0.75 1.00 1.50 2.00 0.2475 0.4810 0.6920 0.8776 1.1798 1.4096 0.2475 0.4810 0.6920 0.8805 5.3064 652.21 G x F x 74. Assume is rational. Let and form the following. Set a positive integer. But, a contradiction. 1 N 1 1 1 N 1 1 N 1 2 . . . 1 N 1 1 1 1 N 1 1 N , a N ! 1 N 1 ! 1 N 2 ! . . . 1 N 1 1 N 1 N 2 . . . < 1 N 1 1 N 1 2 . . . a N ! e 1 1 . . . 1 N ! , e 1 1 1 2! . . . 1 N ! 1 N 1 ! 1 N 2 ! . . . N > q e p q 66. (odd coefficients are zero) a 2 n 1 0 Review Exercises for Chapter 8 2. a n n n 2 1 4. 3.5, 3, . . . Matches (c) a n 4 n 2 : 6. . . . Matches (b) 6, 4, a n 6 2 3 n 1 : (b) 0 2 0.5 1.5 414 Chapter 8 Infinite Series
8. The sequence seems to diverge (oscillates). 1, 0, , 0, 1, 0, . . . 1 sin n 2 : 0 12 2 2 a n sin n 2 10. Converges lim n 1 n 0 12. Diverges lim n n ln n lim n 1 1 n 14. Converges; k 2 n lim n 1 1 2 n n lim k 1 1 k k 1 2 e 1 2 16. Let Assume b c and note that the terms converge as Hence converges. a n n . b n ln b c n ln c b n c n b n ln b b n c n c n ln c b n c n lim n ln y lim n 1 b n c n b n ln b c n ln c ln y ln b n c n n y b n c n 1 n 18. (a) (b) V 5 120,000 0.70 5 $20,168.40 n 1, 2, 3, 4, 5 V n 120,000 0.70 n , k 5 10 15 20 25 0.3917 0.3228 0.3627 0.3344 0.3564 S k 20. (a) (c) The series converges by the Alternating Series Test. (b) 0 12 0 1 k 5 10 15 20 25 0.8333 0.9091 0.9375 0.9524 0.9615 S k 22. (a) (c) The series converges, by the limit comparison test with 1 n 2 . (b) 0 12 0 1 24. Diverges. Geometric series, r 1.82 > 1. 26. Diverges. n th Term Test, lim n a n 2 3 . 28. See Exercise 27. n 0 2 n 2 3 n 4 n 0 2 3 n 4 3 12 30. 1 1 2 3 1 1 2 1 2 1 3 1 3 1 4 . . . 3 1 2 n 0 2 3 n 1 n 1 n 2 n 0 2 3 n n 0 1 n 1 1 n 2 Review Exercises for Chapter 8 415
34. $4,371,379.65 S 39 n 0 32,000 1.055 n 32,000 1 1.055 40 1 1.055 36. See Exercise 86 in Section 8.2. $16,840.32 100 12 0.065 1 0.065 12 120 1 A P 12 r 1 r 12 12 t 1 38. Divergent p -series, p 3 4 < 1 n 1 1 4 n 3 n 1 1 n 3 4 40.

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