thomasET_226348_ism54

thomasET_226348_ism54 - CHAPTER 11 INFINITE SEQUENCES AND...

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CHAPTER 11 INFINITE SEQUENCES AND SERIES 11.1 SEQUENCES 1. a 0, a , a , a " #$% ±" ± " ± ± # œœ ± ± ± 11 2 13 2 14 3 1 4 39 41 6 # ### 2. a 1, a , a , a " "" # œœ œœ 1 1 1 1! ! 2 3! 6 4! 24 3. a 1, a , a , a "# $ % ±± " ± ± ± ± ± " œœœœ ±œœ ± (1 ) ( ) ) ) 1 4 6 1 5 8 17 #$ % & 4. a 2 ( 1) 1, a 2 ( 1) 3, a 2 ( 1) 1, a 2 ( 1) 3 "#$% œ²± œ 5. a , a , a , a # #### """" # 2222 22 & % 6. a , a , a , a " ± " " ±±± " ## œ œœœœœœœ 1 3 2 1 7 2 1 5 24 28 1 6 2 % 7. a 1, a 1 , a , a , a , a , $ % & ' "" " " # # œ œ²œ œ² œ œ ² œ œ 3 3 7 7 15 15 31 63 4 4 8 8 16 32 % a, a ()*" ! œœœ 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 ˆ‰ ˆ ‰ " 3 6 4 4 5 1 0 7 0 5040 40,320 64 a *" ! 362,880 3,628,800 9. a 2, a 1, a , a , a , $ % & # # " œ ±œ œ œ ) ( 2 ) ) ( 1 ) 8 ) ) % " # & " 4 a , a , a , a , a '( ) * " ! """ " " œ 16 3 64 1 8 256 10. a 2, a 1, a , a , a , a , " # $ %&' œ± œ œ œ œ 1( 2) 2( 1) 33 4 5 5 3 34 †† 2 3 " # a ! 749 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 12. a 2, a 1, a , a , a 1, a 2, a 2, a 1, a , a $ % & ' () * " ! # ± ± œ œ œ œ œ œ ˆ " # 1 13. a ( 1) , n 1, 2, 14. a ( 1) , n 1, 2, n n n1 n œ á œ á ² 15. a ( 1) n , n 1, 2, 16. a , n 1, 2, n n n œ á œ œ á ²# ±" # 17. a n 1, n 1, 2, 18. a n 4 , n 1, 2, n n œ ± œá œ ±œá # 19. a 4n 3, n 1, 2, 20. a 4n 2 , n 1, 2, n n œ± œ á 21. a , n 1, 2, 22. a , n 1, 2, n n 1(1 ) n( 1 ) n á œ œ Ú Û œ á ²± # ±²± n 23. lim 2 (0.1) 2 converges (Theorem 5, #4) n Ä_ ²œ Ê n
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694 Chapter 11 Infinite Sequences and Series 24. lim lim 1 1 converges nn Ä_ n( ) (1 ) ±² " ² œ± œ Ê 25. 1 converges n ² ±# # ² ± 2n 2 1n 2 2 œœ œ ² Ê ˆ‰ " " n n 26. diverges 2n 13n 2n 3 ±" ² ± ² È È Š‹ ² _ Ê " " È È n n 27. 5 converges ± ² ± 5n n8 n 5 1 % %$ " % ² Ê n 8 n 28. 0 converges n n3 n 5n 6 (n 3)(n 2) n ±± " ± ± ± # # œ Ê 29. lim (n 1) diverges n n2 n1 (n 1)(n 1) # ²± ²² ² œ _ Ê 30 diverges ² ² ² n 70 4n n 4 $ # _ Ê " # # n 70 n 31. 1 ( 1) does not exist diverges 32. lim ( 1) 1 does not exist diverges ab Ê ² ² Ê n n " n 33. 1 1 converges ˆ ˆ n n n " " " " " ## # # ²œ ± ²œÊ 34. 2 3 6 converges 35. 0 converges ² ± œÊ œÊ "" # ² ²" () 36. 0 converges œ Ê " ²" n n n 37. 2 converges n É É Ê È 2n 2n œ Ê 2 1 ± " n 38. diverges (0.9) 9 0 n _ Ê n 39. lim sin sin sin 1 converges 11 1 # ±œ ± œ œ Ê 40. lim n cos (n ) lim (n )( 1) does not exist diverges 1 œ² Ê n 41. 0 because converges by the Sandwich Theorem for sequences n sin n sin n n n Ÿ Ÿ Ê 42. 0 because 0 converges by the Sandwich Theorem for sequences n sin n sin n # " n œŸ Ÿ Ê 43. 0 converges (using l'Hopital's rule) ^ n ln 2 " Ê 44. diverges (using l'Hopital's rule) ^ n n 33 l n 3 n 6 n 6 3( ln 3 ) 3 ) $# #$ œ œœœ _ Ê 45. 0 converges n n ln (n ) n 1 ± ± È È œ œ Ê " " # " n 2 n n È È
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Section 11.1 Sequences 695 46. lim 1 converges nn Ä_ ln n ln 2n œœ Ê ˆ‰ " n 2 2n 47. lim 8 1 converges (Theorem 5, #3) n 1n Î œÊ 48. lim (0.03) 1 converges (Theorem 5, #3) n Î 49. 1 e converges (Theorem 5, #5) n ±œ Ê 7 n n ( 50. 1 1 e converges (Theorem 5, #5) ’“ ²œ ± œ Ê " ±" ±" () n n 51. 10n lim 10 n 1 1 1 converges (Theorem 5, #3 and #2) È n œ Ê ÎÎ †† 52. n n 1 1 converges (Theorem 5, #2) È È n n # # # œ Ê 53. 1 converges (Theorem 5, #3 and #2) n 3 n1 lim 3 lim n Î " œ Ê n n Ä_ Ä_ Î Î 54.
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thomasET_226348_ism54 - CHAPTER 11 INFINITE SEQUENCES AND...

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