Chapter9 - CHAPTER Infinite Series 9 Section 9.1 Sequences...

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CHAPTER 9 Infinite Series Section 9.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 437 Section 9.2 Series and Convergence . . . . . . . . . . . . . . . . . . . 444 Section 9.3 The Integral Test and p -Series . . . . . . . . . . . . . . . . 451 Section 9.4 Comparisons of Series . . . . . . . . . . . . . . . . . . . . 457 Section 9.5 Alternating Series . . . . . . . . . . . . . . . . . . . . . . 460 Section 9.6 The Ratio and Root Tests . . . . . . . . . . . . . . . . . . 465 Section 9.7 Taylor Polynomials and Approximations . . . . . . . . . . 472 Section 9.8 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . 477 Section 9.9 Representation of Functions by Power Series . . . . . . . 483 Section 9.10 Taylor and Maclaurin Series . . . . . . . . . . . . . . . . 487 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
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CHAPTER 9 Infinite Series Section 9.1 Sequences 437 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. 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 13. 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 15. decreases to 0; matches (f). a n 5 8 n 1 1 , a 1 5 4, a 2 5 8 3 , 17. decreases to 0; matches (e). a n 5 4 s 0.5 d n 2 1 , a 1 5 4, a 2 5 2, 19. etc.; matches (d). a 3 52 1, a n s 2 1 d n , a 1 1, a 2 5 1, 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 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 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. Add 3 to preceding 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 27. a 6 5 2 s 80 d 5 160 a 5 5 2 s 40 d 5 80 a 1 5 5 a n 1 1 5 2 a n ,
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438 Chapter 9 Infinite Series 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 29. Multiply the preceding term by 2 1 2 . a 6 5 3 s 2 2 d 5 52 3 32 a 5 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. s n 1 1 d ! n ! 5 n ! s n 1 1 d n ! 5 n 1 1 39. lim n ` 2 n ! n 2 1 1 5 lim n ` 2 ! 1 1 s 1 y n 2 d 5 2 1 5 2 41. lim n ` sin 1 1 n 2 5 0 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. Thus, converges. lim n ` a n 5 0, 5 1 2 n ? 3 2 n ? 5 2 n . . . 2 n 2 1 2 n < 1 2 n a n 5 1 ? 3 ? 5 . . .
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Chapter9 - CHAPTER Infinite Series 9 Section 9.1 Sequences...

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