Chapter 11 - CHAPTER 11 INFINITE SEQUENCES AND SERIES 11.1...

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CHAPTER 11 INFINITE SEQUENCES AND SERIES 11.1 SEQUENCES 1. a 0, a , a , a " # $ % " " # œ œ œ œ œ œ œ œ 1 1 2 1 3 2 1 4 3 1 4 3 9 4 16 2. a 1, a , a , a " # $ % " " # œ œ œ œ œ œ œ œ 1 1 1 1 1 1! ! 2 3! 6 4! 24 3. a 1, a , a , a " # $ % " # " " " œ œ œ œ œ œ œ œ ( 1) ( ) ( 1) ( 1) 1 4 1 3 6 1 5 8 1 7 4. a 2 ( 1) 1, a 2 ( 1) 3, a 2 ( 1) 1, a 2 ( 1) 3 " # $ % " # $ % œ œ œ œ œ œ œ œ 5. a , a , a , a " # $ % # # # # # " " " " # œ œ œ œ œ œ œ œ 2 2 2 2 2 2 6. a , a , a , a " # $ % " " " # # œ œ œ œ œ œ œ œ 2 2 1 3 2 1 7 2 15 2 4 2 8 16 2 7. a 1, a 1 , a , a , a , a , " # $ % & ' " " " " # # # # # # œ œ œ œ œ œ œ œ œ œ 3 3 7 7 15 15 31 63 4 4 8 8 16 32 a , a , a , a ( ) * "! œ œ œ œ 127 255 511 1023 64 128 256 512 8. a 1, a , a , a , a , a , a , a , " # $ % & ' ( ) " " " " " " " # # # # œ œ œ œ œ œ œ œ œ œ œ ˆ ˆ ‰ ˆ 3 6 4 4 5 1 0 7 0 5040 40,320 6 4 a , a * "! " " œ œ 362,880 3,628,800 9. a 2, a 1, a , a , a , " # $ % & # # # # " " " œ œ œ œ œ œ œ œ œ ( 1) (2) ( 1) (1) 2 4 8 ( 1) ( 1) ˆ ˆ 4 a , a , a , a , a ' ( ) * "! " " " " " # # œ œ œ œ œ 16 3 64 1 8 256 10. a 2, a 1, a , a , a , a , " # $ % & ' # # " " œ œ œ œ œ œ œ œ œ œ 1 ( 2) 2 ( 1) 3 3 4 5 5 3 2 2 3 4 ˆ ˆ 2 3 a , a , a , a ( ) * "! " " œ œ œ œ 2 2 7 4 9 5 11. a 1, a 1, a 1 1 2, a 2 1 3, a 3 2 5, a 8, a 13, a 21, a 34, a 55 " # $ % & ' ( ) * "! œ œ œ œ œ œ œ œ œ œ œ œ œ 12. a 2, a 1, a , a , a 1, a 2, a 2, a 1, a , a " # $ % & ' ( ) * "! " " " " # # # # œ œ œ œ œ œ œ œ œ œ œ œ ˆ ˆ ‰ ˆ 1 13. a ( 1) , n 1, 2, 14. a ( 1) , n 1, 2, n n n 1 n œ œ á œ œ á 15. a ( 1) n , n 1, 2, 16. a , n 1, 2, n n n 1 ( ) n œ œ á œ œ á # " n 1 17. a n 1, n 1, 2, 18. a n 4 , n 1, 2, n n œ œ á œ œ á # 19. a 4n 3, n 1, 2, 20. a 4n 2 , n 1, 2, n n œ œ á œ œ á 21. a , n 1, 2, 22. a , n 1, 2, n n 1 ( 1) n ( 1) n œ œ á œ œ Ú Û œ á # # # n 1 n ˆ ‰ 23. lim 2 (0.1) 2 converges (Theorem 5, #4) n Ä _ œ Ê n
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698 Chapter 11 Infinite Sequences and Series 24. lim lim 1 1 converges n n Ä _ Ä _ n ( ) ( 1) n n " n n œ œ Ê 25. lim lim lim 1 converges n n n Ä _ Ä _ Ä _ " # # 2n 2 1 n 2 2 œ œ œ Ê ˆ ‰ ˆ ‰ n n 26. lim lim diverges n n Ä _ Ä _ 2n 1 3 n 2 n 3 " È È Š Š œ œ _ Ê n n 27. lim lim 5 converges n n Ä _ Ä _ " 5n n 8n 5 1 œ œ Ê Š ˆ ‰ n 8 n 28. lim lim lim 0 converges n n n Ä _ Ä _ Ä _ n 3 n 3 n 5n 6 (n 3)(n 2) n " # œ œ œ Ê 29. lim lim lim (n 1) diverges n n n Ä _ Ä _ Ä _ n 2n 1 n 1 n 1 (n 1)(n 1) œ œ œ _ Ê 30 lim lim diverges n n Ä _ Ä _ " n 70 4n n 4 œ œ _ Ê Š Š n 70 n 31. lim 1 ( 1) does not exist diverges 32. lim ( 1) 1 does not exist diverges n n Ä _ Ä _ a b ˆ Ê Ê n n " n 33. lim 1 lim 1 converges n n Ä _ Ä _ ˆ ‰ ˆ ˆ ‰ ˆ n n n n n " " " " " " # # # # œ œ Ê 34. lim 2 3 6 converges 35. lim 0 converges n n Ä _ Ä _ ˆ ‰ ˆ œ Ê œ Ê " " # # # " n n n 1 ( ) n 1 36. lim lim 0 converges n n Ä _ Ä _ ˆ œ œ Ê " # # " n ( ) n n 37. lim lim lim 2 converges n n n Ä _ Ä _ Ä _ É É Ê Š È 2n 2n n 1 n 1 œ œ œ Ê 2 1 n 38. lim lim diverges n n Ä _ Ä _ " " (0.9) 9 0 n œ œ _ Ê ˆ n 39. lim sin sin lim sin 1 converges n n Ä _ Ä _ ˆ ˆ Š 1 1 1 # # # " " œ œ œ Ê n n 40. lim n cos (n ) lim (n )( 1) does not exist diverges n n Ä _ Ä _ 1 1 1 œ Ê n 41. lim 0 because converges by the Sandwich Theorem for sequences n Ä _ sin n sin n n n n n œ Ÿ Ÿ Ê " " 42. lim 0 because 0 converges by the Sandwich Theorem for sequences n Ä _ sin n sin n # # # " n n n œ Ÿ Ÿ Ê 43. lim lim 0 converges (using l'Hopital's rule) ^ n n Ä _ Ä _ n ln 2 # # " n n œ œ Ê 44. lim lim lim lim diverges (using l'Hopital's rule) ^ n n n n Ä _ Ä _ Ä _ Ä _ 3 3 ln 3 n 3n 6n 6 3 (ln 3) 3 (ln 3) n n n n œ œ œ œ _ Ê 45. lim lim
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