# ODDREV08 - R eview Exercises for Chapter 8 167 Review...

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Review Exercises for Chapter 8 1. a n 1 n ! 3. 6, 5, 4.67, . . . Matches (a) a n 4 2 n : 5. 10, 3, . . . Matches (d) a n 10 0.3 n 1 : 7. The sequence seems to converge to 5. lim n 5 2 n 5 lim n a n lim n 5 n 2 n 12 0 0 8 a n 5 n 2 n 13. Converges lim n 1 n 1 n 0 lim n n 1 n lim n n 1 n n 1 n n 1 n 15. Converges lim n sin n n 0 17. (a) (b) A 40 8218.10 A 4 5254.73 A 8 5522.43 A 3 5189.85 A 7 5454.25 A 2 5125.78 A 6 5386.92 A 1 5062.50 A 5 5320.41 n 1, 2, 3 A n 5000 1 0.05 4 n 5000 1.0125 n k 5 10 15 20 25 13.2 113.3 873.8 6448.5 50,500.3 S k 19. (a) (c) The series diverges geometric r 3 2 > 1 (b) 12 0 10 120 k 5 10 15 20 25 0.4597 0.4597 0.4597 0.4597 0.4597 S k 21. (a) (c) The series converges by the Alternating Series Test. (b) 12 0 0.1 1 23. Converges. Geometric series, r 0.82, r < 1. 25. Diverges. n th Term Test. lim n a n 0. 9. Converges lim n n 1 n 2 0 11. lim n n 3 n 2 1 Review Exercises for Chapter 8 167
27. Geometric series with and S a 1 r 1 1 2 3 1 1 3 3 r 2 3 . a 1 n 0 2 3 n 31. n 0 0.09 0.01 n 0.09 1 0.01 1 11 0.09 0.09 0.0009 0.000009 . . . 0.09 1 0.01 0.0001 . . . 33. meters 8 n 0 16 0.7 n 8 16 1 0.7 45 1 3 D 8 16 0.7 16 0.7 2 . . . 16 0.7 n . . . D 2 0.7 8 0.7 8 16 0.7 D 1 8 35. See Exercise 86 in Section 8.2. \$5087.14 200 e 0.06 2 1 e 0.06 12 1 A P e rt 1 e r 12 1 37. By the Integral Test, the series converges. 0 1 9 1 9 1 x 4 ln x dx lim b ln x 3 x 3 1 9 x 3 b 1 29. 1 1 1 2 1 1 1 3 2 3 2 1 2 n 0 1 2 n 1 3 n n 0 1 2 n n 0 1 3 n 39. Since the second series is a divergent p -series while the first series is a convergent p -series, the difference diverges. n 1 1 n 2 1 n n 1 1 n 2 n 1 1 n 41. By a limit comparison test with the convergent p -series the series converges. n 1 1 n 3 2 , lim n 1 n 3 2 n 1 n 3 2 lim n n 3 2 n 3 2 n 1 n 1 1 n 3 2 n 43. Since diverges (harmonic series), so does the original series. n 1 1 2 n 1 2 n 1 1 n 3 2 5 4 . . . 2 n 1 2 n 2 1 2 n > 1 2 n a n 1 3 5 . . . 2 n 1 2 4 6 . . . 2 n n 1 1 3 5 . . . 2 n 1 2 4 6 . . . 2 n 45. Converges by the Alternating Series Test (Conditional convergence) 47. Diverges by the n th Term Test 49. By the Ratio Test, the series converges. 0 1 0 < 1 lim n 1 e 2 n 1 n 1 n lim n e n 2 n 1 e n 2 2 n 1 n lim n a n 1 a n lim n n 1 e n 1 2 e n 2 n n 1 n e n 2 51.