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09 - CHAPTER Infinite Series Section 9.1 Section 9.2...

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C H A P T E R 9 Infinite Series Section 9.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Section 9.2 Series and Convergence . . . . . . . . . . . . . . . . . . . 247 Section 9.3 The Integral Test and p -Series . . . . . . . . . . . . . . . . 260 Section 9.4 Comparisons of Series . . . . . . . . . . . . . . . . . . . . 270 Section 9.5 Alternating Series . . . . . . . . . . . . . . . . . . . . . . 277 Section 9.6 The Ratio and Root Tests . . . . . . . . . . . . . . . . . . 286 Section 9.7 Taylor Polynomials and Approximations . . . . . . . . . . 298 Section 9.8 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . 308 Section 9.9 Representation of Functions by Power Series . . . . . . . 321 Section 9.10 Taylor and Maclaurin Series . . . . . . . . . . . . . . . . 329 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
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C H A P T E R 9 Infinite Series Section 9.1 Sequences 233 1. a 5 2 5 32 a 4 2 4 16 a 3 2 3 8 a 2 2 2 4 a 1 2 1 2 a n 2 n 2. a 5 3 5 5! 243 120 81 40 a 4 3 4 4! 81 24 27 8 a 3 3 3 3! 27 6 9 2 a 2 3 2 2! 9 2 a 1 3 1! 3 a n 3 n n ! 3. a 5 1 2 5 1 32 a 4 1 2 4 1 16 a 3 1 2 3 1 8 a 2 1 2 2 1 4 a 1 1 2 1 1 2 a n 1 2 n 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 5. a 5 sin 5 2 1 a 4 sin 2 0 a 3 sin 3 2 1 a 2 sin 0 a 1 sin 2 1 a n sin n 2 6. 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 7. a 5 1 15 5 2 1 25 a 4 1 10 4 2 1 16 a 3 1 6 3 2 1 9 a 2 1 3 2 2 1 4 a 1 1 1 1 2 1 a n 1 n n 1 2 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 9. a 5 5 1 5 1 25 121 25 a 4 5 1 4 1 16 77 16 a 3 5 1 3 1 9 43 9 a 2 5 1 2 1 4 19 4 a 1 5 1 1 5 a n 5 1 n 1 n 2
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234 Chapter 9 Infinite Series 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 11. 2 10 1 18 a 5 2 a 4 1 2 6 1 10 a 4 2 a 3 1 2 4 1 6 a 3 2 a 2 1 2 3 1 4 a 2 2 a 1 1 a 1 3, a k 1 2 a k 1 12. 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 13. a 5 1 2 a 4 1 2 4 2 a 4 1 2 a 3 1 2 8 4 a 3 1 2 a 2 1 2 16 8 a 2 1 2 a 1 1 2 32 16 a 1 32, a k 1 1 2 a k 14. 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 15. decreases to 0; matches (f). a n 8 n 1 , a 1 4, a 2 8 3 , 16. increases towards 8; matches (a). a n 8 n n 1 , a 1 4, a 2 16 3 , 17. decreases to 0; matches (e). a n 4 0.5 n 1 , a 1 4, a 2 2, 18. eventually approaches 0; matches (b). a n 4 n n ! , a 1 4, a 2 8, 19. etc.; matches (d). a 3 1, a n 1 n , a 1 1, a 2 1, 20. etc.; matches (c). a 3 1 3 , a n 1 n n , a 1 1, a 2 1 2 , 21. a n 2 3 n , n 1, . . . , 10 1 1 12 8 22. n 1, . . . , 10 a n 2 4 n , 1 12 3 4 23. a n 16 0.5 n 1 , n 1, . . . , 10 12 1 10 18 24. a n 2 n n 1 , n 1, 2, . . . , 10 12 1 1 3 25. Add 3 to preceding term. a 6 3 6 1 17 a 5 3 5 1 14 a n 3 n 1 26. a 6 6 6 2 6 a 5 5 6 2 11 2 a n n 6 2
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Section 9.1 Sequences 235 27. a 6 2 80 160 a 5 2 40 80 a 1 5 a n 1 2 a n , 28. a 5 1 16 , a 6 1 32 a n 1 2 a n 1 , a 1 1 29. Multiply the preceding term by 1 2 . a 6 3 2 5 3 32 a 5 3 2 4 3 16 a n 3 2 n 1 30. a 5 81 16 , a 6 243 32 a n 3 2 a n 1 , a 1 1 31. 9 10 90 10! 8! 8! 9 10 8! 32. 24 25 600 25! 23! 23! 24 25 23! 33. n 1 ! n ! n ! n 1 n ! n 1 34. n 1 n 2 n 2 ! n ! n ! n 1 n 2 n ! 35. 1 2 n 2 n 1 2 n 1 ! 2 n 1 ! 2 n 1 ! 2 n 1 ! 2 n 2 n 1 36. 2 n 1 2 n 2 2 n 2 ! 2 n ! 2 n ! 2 n 1 2 n 2 2 n ! 37. lim n 5 n 2 n 2 2 5 38. lim n 5 1 n 2 5 0 5 39. lim n 2 n n 2 1 lim n 2 1 1 n 2 2 1 2 40. lim n 5 n n 2 4 lim n 5 1 4 n 2 5 1 5 41. lim n sin 1 n 0 42. lim n cos 2 n 1 43.
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