MTH 201 Homework 3 Solution, Fall 2016

MTH 201. Homework III Solution. Section 10.1: !"#$%&'()%$*+,-*./%000%1*2345*6)%784*9,.%&:;%&'((% % % % % % % % % % % % % % % % 35. lim 1 ( 1) does not exist diverges 36. lim ( 1) 1 does not exist diverges n n Ä _ Ä _ a b ˆ ‰ � � Ê � � Ê n n " n 40. lim lim 0 converges n n Ä _ Ä _ ˆ � ‰ œ œ Ê " # # �" n ( )n n 41. 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 43. lim sin sin lim sin 1 converges n n Ä _ Ä _ ˆ ‰ ˆ ‰ Š ‹ 1 1 1 # # # " " � œ � œ œ Ê n n 44. lim n cos (n ) lim (n )( 1) does not exist diverges n n Ä _ Ä _ 1 1 1 œ � Ê n 45. lim 0 because converges by the Sandwich Theorem for sequences n Ä _ sin n sin n n n n n œ � Ÿ Ÿ Ê " " 47. lim lim 0 converges (using l'Hopital's rule) ^ n n Ä _ Ä _ n ln 2 # # " n n œ œ Ê 50. lim lim 1 converges n n Ä _ Ä _ ln n ln 2n œ œ Ê ˆ ‰ ˆ ‰ n 2 2n 51. lim 8 1 converges (Theorem 5, #3) n Ä _ 1 n œ Ê 52. lim (0.03) 1 converges (Theorem 5, #3) n Ä _ 1 n œ Ê 53. lim 1 e converges (Theorem 5, #5) n Ä _ ˆ ‰ � œ Ê 7 n n ( 55. lim 10n lim 10 n 1 1 1 converges (Theorem 5, #3 and #2) n n Ä _ Ä _ È n œ œ œ Ê 1 n 1 n Î Î † † 56. lim n lim n 1 1 converges (Theorem 5, #2) n n Ä _ Ä _ È ˆ ‰ È n n # # # œ œ œ Ê 57. lim 1 converges (Theorem 5, #3 and #2) n Ä _ ˆ ‰ 3 n 1 1 n lim 3 lim n Î " œ œ œ Ê n n 1 n 1 n 59. lim diverges (Theorem 5, #2) n Ä _ ln n n lim n lim ln n 1 1 n 1 n œ œ œ _ Ê n n _ 63. lim lim lim 0 and 0 lim 0 converges n n n n Ä _ Ä _ Ä _ Ä _ n! n! n! n n n n n n n n n 2 3 (n 1)(n) n n n œ Ÿ œ Ê œ Ê " â � â " † † † † † ˆ ‰ 67. lim lim exp ln lim exp e converges n n n Ä _ Ä _ Ä _ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ " " " � ÎÐ Ñ �" n ln n n ln n 1 ln n ln 1 ln n œ œ œ Ê Section 10.2: % % % % % 70. lim lim exp n ln lim exp lim exp n n n n Ä _ Ä _ Ä _ Ä _ ˆ ‰ ˆ ‰ ˆ ‰ Š ‹ � � n n n 1 n 1 n ln n ln (n 1) � � � � œ œ œ ˆ ‰ Š ‹ n n n 1 n � � lim exp e converges œ � œ Ê n Ä _ Š ‹ n n(n 1) � �" 71. lim lim x x lim exp ln x lim exp n n n n Ä _ Ä _ Ä _ Ä _ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ Š ‹ x 2n 1 n 1 n n 1 n 1 n 1 n ln (2n 1) n � # � # � Î " " " Î � � œ œ œ x lim exp xe x, x 0 converges œ œ œ � Ê n Ä _ ˆ ‰ � � ! 2 2n 1 72. lim 1 lim exp n ln 1 lim exp lim exp n n n n Ä _ Ä _ Ä _ Ä _ ˆ ‰ ˆ ‰ ˆ ‰ � � – — � œ � œ œ " " n n n ln 1 1 Š ‹ ‚ ˆ ‰ Š ‹ Š ‹ Š ‹ � � � n n 2 n n n lim exp e 1 converges œ œ œ Ê n Ä _ ˆ ‰ � � ! 2n n 1 73. lim lim 0 converges (Theorem 5, #6) n n Ä _ Ä _ 3 6 36 2 n! n! n n n n† † œ œ Ê 74. lim lim lim 0 converg n n n Ä _ Ä _ Ä _ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ 10 12 10 120 11 11 11 121 n n n n 9 11 12 9 12 11 108 10 12 11 10 11 12 110 n n n n n n n � � � œ œ œ Ê 1 es (Theorem 5, #4) 75. lim tanh n lim lim lim lim 1 converges n n n n n Ä _ Ä _ Ä _ Ä _ Ä _ œ œ œ œ " œ Ê e e e 2e e e e 1 2e n n 2n 2n n n 2n 2n � � " � � 76. lim sinh (ln n) lim lim diverges n n n Ä _ Ä _ Ä _ œ œ œ _ Ê e e 2 n ln n ln n n � � # ˆ ‰ 77. lim lim lim lim converges n n n n Ä _ Ä _ Ä _ Ä _ n sin 2n 1 ˆ ‰ n � # " œ œ œ œ Ê sin cos cos ˆ ‰ ˆ ‰ Š ‹ Š ‹ ˆ ‰ ˆ ‰ Š ‹ ˆ ‰ n n 2 2 2 n n n n n n 2 n � � � � � �# � 78. lim n 1 cos lim lim lim sin 0 converges n n n n Ä _ Ä _ Ä _ Ä _ ˆ ‰ ˆ ‰ � œ œ œ œ Ê " " n n ˆ ‰ ˆ ‰ � ‘ ˆ ‰ Š ‹ Š ‹ " � cos sin n n n n n 79. lim n sin lim lim lim cos cos 0 1 converges n n n n Ä _ Ä _ Ä _ Ä _ È Š ‹ Š ‹ 1 1 n n sin cos È È Š ‹ Š ‹Š ‹ œ œ œ œ œ Ê 1 1 1 n n 1 1 n 2n3 2 2n3 2 � � 80. lim 3 5 lim exp ln 3 5 lim exp lim exp n n n n Ä _ Ä _ Ä _ Ä _ a b a b ’ “ ’ “ – — n n n n 1 n 1 n ln 3 5 n 1 � œ � œ œ Î Î � a b n n 3 ln 3 5 ln 5 n n 3 5 n n lim exp lim exp exp ln 5 5 œ œ œ œ n n Ä _ Ä _ ’ “ ’ “ a b Š ‹ ˆ ‰ ˆ ‰ ˆ ‰ 3n 5n 3 3 n n 5 5 n 3 5 n ln 3 ln 5 1 1 ln 3 ln 5 � � � � 81. lim tan n converges 82. lim tan n 0 0 converges n n Ä _ Ä _ �" �" # # " œ Ê œ œ Ê 1 1 Èn † Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 74. lim lim lim 0 converg n n n Ä _ Ä _ Ä _ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ 10 12 10 120 11 11 11 121 n n n n 9 11 12 9 12 11 108 10 12 11 10 11 12 110 n n n n n n n � � � œ œ œ Ê 1 es (Theorem 5, #4) 75. lim tanh n lim lim lim lim 1 converges n n n n n Ä _ Ä _ Ä _ Ä _ Ä _ œ œ œ œ " œ Ê e e e 2e e e e 1 2e n n 2n 2n n n 2n 2n � � " � � 76. lim sinh (ln n) lim lim diverges n n n Ä _ Ä _ Ä _ œ œ œ _ Ê e e 2 n ln n ln n n � � # ˆ ‰ 77. lim lim lim lim converges n n n n Ä _ Ä _ Ä _ Ä _ n sin 2n 1 ˆ ‰ n � # " œ œ œ œ Ê sin cos cos ˆ ‰ ˆ ‰ Š ‹ Š ‹ ˆ ‰ ˆ ‰ Š ‹ ˆ ‰ n n 2 2 2 n n n n n n 2 n � � � � � �# � 78. lim n 1 cos lim lim lim sin 0 converges n n n n Ä _ Ä _ Ä _ Ä _ ˆ ‰ ˆ ‰ � œ œ œ œ Ê " " n n ˆ ‰ ˆ ‰ � ‘ ˆ ‰ Š ‹ Š ‹ " � cos sin n n n n n 79. lim n sin lim lim lim cos cos 0 1 converges n n n n Ä _ Ä _ Ä _ Ä _ È Š ‹ Š ‹ 1 1 n n sin cos È È Š ‹ Š ‹Š ‹ œ œ œ œ œ Ê 1 1 1 n n 1 1 n 2n3 2 2n3 2 � � 80. lim 3 5 lim exp ln 3 5 lim exp lim exp n n n n Ä _ Ä _ Ä _ Ä _ a b a b ’ “ ’ “ – — n n n n 1 n 1 n ln 3 5 n 1 � œ � œ œ Î Î � a b n n 3 ln 3 5 ln 5 n n 3 5 n n lim exp lim exp exp ln 5 5 œ œ œ œ n n Ä _ Ä _ ’ “ ’ “ a b Š ‹ ˆ ‰ ˆ ‰ ˆ ‰ 3n 5n 3 3 n n 5 5 n 3 5 n ln 3 ln 5 1 1 ln 3 ln 5 � � � � 81. lim tan n converges 82. lim tan n 0 0 converges n n Ä _ Ä _ �" �" # # " œ Ê œ œ Ê 1 1 Èn † Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 76. lim sinh (ln n) lim lim diverges n n n Ä _ Ä _ Ä _ œ œ œ _ Ê 2 n # 77. lim lim lim lim converges n n n n Ä _ Ä _ Ä _ Ä _ n sin 2n 1 ˆ ‰ n � # " œ œ œ œ Ê sin cos cos ˆ ‰ ˆ ‰ Š ‹ Š ‹ ˆ ‰ ˆ ‰ Š ‹ ˆ ‰ n n 2 2 2 n n n n n n 2 n � � � � � �# � 78. lim n 1 cos lim lim lim sin 0 converges n n n n Ä _ Ä _ Ä _ Ä _ ˆ ‰ ˆ ‰ � œ œ œ œ Ê " " n n ˆ ‰ ˆ ‰ � ‘ ˆ ‰ Š ‹ Š ‹ " � cos sin n n n n n 79. lim n sin lim lim lim cos cos 0 1 converges n n n n Ä _ Ä _ Ä _ Ä _ È Š ‹ Š ‹ 1 1 n n sin cos È È Š ‹ Š ‹Š ‹ œ œ œ œ œ Ê 1 1 1 n n 1 1 n 2n3 2 2n3 2 � � 80. lim 3 5 lim exp ln 3 5 lim exp lim exp n n n n Ä _ Ä _ Ä _ Ä _ a b a b ’ “ ’ “ – — n n n n 1 n 1 n ln 3 5 n 1 � œ � œ œ Î Î � a b n n 3 ln 3 5 ln 5 n n 3 5 n n lim exp lim exp exp ln 5 5 œ œ œ œ n n Ä _ Ä _ ’ “ ’ “ a b Š ‹ ˆ ‰ ˆ ‰ ˆ ‰ 3n 5n 3 3 n n 5 5 n 3 5 n ln 3 ln 5 1 1 ln 3 ln 5 � � � � 81. lim tan n converges 82. lim tan n 0 0 converges n n Ä _ Ä _ �" �" # # " œ Ê œ œ Ê 1 1 Èn † Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 83. lim lim 0 converges (Theorem 5, #4) n n Ä _ Ä _ ˆ ‰ Šˆ ‰ ‹ Š ‹ " " " " 3 3 n n 2 2 n � œ � œ Ê È È n 84. lim n n lim exp lim exp e 1 converges n n n Ä _ Ä _ Ä _ È ’ “ ˆ ‰ n # � � � ! � œ œ œ œ Ê ln n n n n n 2n 1 a b 85. lim lim lim lim 0 converges n n n n Ä _ Ä _ Ä _ Ä _ (ln n) 200 (ln n) 200 199 (ln n) n n n n 200! œ œ œ á œ œ Ê † 86. lim lim lim lim lim 0 converges n n n n n Ä _ Ä _ Ä _ Ä _ Ä _ (ln n) 10(ln n) 80(ln n) n n n n 3840 È È È È œ œ œ œ á œ œ Ê – — Š ‹ Š ‹ 5(ln n) n n 87. lim n n n lim n n n lim lim n n n n Ä _ Ä _ Ä _ Ä _ Š ‹ Š ‹ Š ‹ È È � � œ � � œ œ # # � � � � � � " � � n n n n n n n n n n 1 1 È È È É n converges œ Ê " # 88. lim lim lim n n n Ä _ Ä _ Ä _ " " � � � � � � � � � � � � � � � � � È È È È È È È È È È n 1 n n n 1 n n n 1 n n n 1 n n n 1 n n 1 n œ œ Š ‹ Š ‹ lim 2 converges œ œ � Ê n Ä _ É É ˆ ‰ 1 1 1 � � � � � n n n 89. lim dx lim lim 0 converges (Theorem 5, #1) n n n Ä _ Ä _ Ä _ " " " n x n n ln n ' 1 n œ œ œ Ê 90. lim dx lim lim 1 if p 1 converges n n n Ä _ Ä _ Ä _ ' 1 n n 1 " " " " " " � � � x 1 p x 1 p n p 1 p p 1 p 1 œ œ � œ � Ê ’ “ ˆ ‰ 91. Since a converges lim a L lim a lim L L 1 L 72 L L 72 0 n n n 1 72 72 1 a 1 L 2 Ê œ Ê œ Ê œ Ê � œ Ê � � œ n n n Ä _ Ä _ Ä _ � � � n a b L 9 or L 8; since a 0 for n 1 L 8 Ê œ � œ � Ê œ n 92. Since a converges lim a L lim a lim L L L 2 L 6 L L 6 0 n n n 1 a 6 a 2 L 2 L 6 2 Ê œ Ê œ Ê œ Ê � œ � Ê � � œ n n n Ä _ Ä _ Ä _ � � � � � n n a b L 3 or L 2; since a 0 for n 2 L 2 Ê œ � œ � Ê œ n 93. Since a converges lim a L lim a lim 8 2a L 8 2L L 2L 8 0 L 2 n n n 1 n 2 Ê œ Ê œ � Ê œ � Ê � � œ Ê œ � n n n Ä _ Ä _ Ä _ � È È or L 4; since a 0 for n 3 L 4 œ � Ê œ n 94. Since a converges lim a L lim a lim 8 2a L 8 2L L 2L 8 0 L 2 n n n 1 n 2 Ê œ Ê œ � Ê œ � Ê � � œ Ê œ � n n n Ä _ Ä _ Ä _ � È È or L 4; since a 0 for n 2 L 4 œ � Ê œ n 95. Since a converges lim a L lim a lim 5a L 5L L 5L 0 L 0 or L 5; since n n n 1 n 2 Ê œ Ê œ Ê œ Ê � œ Ê œ œ n n n Ä _ Ä _ Ä _ � È È a 0 for n 1 L 5 n � Ê œ 96. Since a converges lim a L lim a lim 12 a L 12 L L 25L 144 0 n n n 1 n 2 Ê œ Ê œ � Ê œ � Ê � � œ n n n Ä _ Ä _ Ä _ � ˆ ‰ È Š ‹ È L 9 or L 16; since 12 a 12 for n 1 L 9 Ê œ œ � � Ê œ È n 97. a 2 , n 1, a 2. Since a converges lim a L lim a lim 2 L 2 n 1 1 n n n 1 1 1 1 a a L � � œ � œ Ê œ Ê œ � Ê œ � n n n n n Ä _ Ä _ Ä _Š ‹ L 2L 1 0 L 1 2; since a 0 for n 1 L 1 2 Ê � � œ Ê œ „ � Ê œ � 2 n È È Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 578 Chapter 10 Infinite Sequences and Series 6. s 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 n(n 1) n n 1 2 2 3 3 4 n 1 n n n 1 n 1 n � � � � � œ � Ê œ � � � � � � á � � � � œ � ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ lim s 5 Ê œ n Ä _ n 7. 1 , the sum of this geometric series is � � � � á œ œ " " " " " � � � 4 16 64 5 1 1 4 ˆ ‰ ˆ ‰ 4 4 8. , the sum of this geometric series is " " " " � # 16 64 256 1 1 � � � á œ ˆ ‰ ˆ ‰ 16 4 9. , the sum of this geometric series is 7 7 7 7 4 16 64 3 1 � � � á œ ˆ ‰ ˆ ‰ 7 4 4 � 10. 5 , the sum of this geometric series is 4 � � � � á œ 5 5 5 5 4 16 64 1 � � ˆ ‰ 4 11. (5 1) , is the sum of two geometric series; the sum is � � � � � � � � á ˆ ‰ ˆ ‰ ˆ ‰ 5 5 5 3 4 9 8 7 # # " " " 10 5 3 23 1 1 � � " # # ˆ ‰ ˆ ‰ � œ � œ 3 12. (5 1) , is the difference of two geometric series; the sum is � � � � � � � � á ˆ ‰ ˆ ‰ ˆ ‰ 5 5 5 3 4 9 8 7 # # " " " 10 5 3 17 1 1 � � " # # ˆ ‰ ˆ ‰ � œ � œ 3 13. (1 1) , is the sum of two geometric series; the sum is � � � � � � � � á ˆ ‰ ˆ ‰ ˆ ‰ 1 1 1 5 4 25 8 1 5 # # " " " 2 1 5 17 1 1 6 6 � � " ˆ ‰ ˆ ‰ � œ � œ 5 14. 2 2 1 ; the sum of this geometric series is 2 � � � � á œ � � � � á œ 4 8 16 2 4 8 10 5 25 125 5 25 125 3 1 ˆ ‰ Š ‹ " � ˆ ‰ 2 5 15. Series is geometric with r 1 Converges to œ Ê � Ê œ 2 2 1 5 5 5 3 1 ¹ ¹ � 2 5 16. Series is geometric with r 3 3 1 Diverges œ � Ê � � Ê ¹ ¹ 17. Series is geometric with r 1 Converges to œ Ê � Ê œ 1 1 1 8 8 7 1 ¹ ¹ 1 8 1 8 � 18. Series is geometric with r 1 Converges to œ � Ê � � Ê œ � 2 2 2 3 3 5 1 ¹ ¹ � � � 2 3 2 3 ˆ ‰ 19. 0.23 20. 0.234 œ œ œ œ œ œ ! ! ˆ ‰ ˆ ‰ _ _ œ œ n 0 n 0 23 23 234 234 100 10 99 1000 10 999 n n 1 1 " " � � Š ‹ Š ‹ ˆ ‰ Š ‹ 23 234 100 1000 1000 100 578 Chapter 10 Infinite Sequences and Series 6. s 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 n(n 1) n n 1 2 2 3 3 4 n 1 n n n 1 n 1 n � � � � � œ � Ê œ � � � � � � á � � � � œ � ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ lim s 5 Ê œ n Ä _ n 7. 1 , the sum of this geometric series is � � � � á œ œ " " " " " � � � 4 16 64 5 1 1 4 ˆ ‰ ˆ ‰ 4 4 8. , the sum of this geometric series is " " " " � # 16 64 256 1 1 � � � á œ ˆ ‰ ˆ ‰ 16 4 9. , the sum of this geometric series is 7 7 7 7 4 16 64 3 1 � � � á œ ˆ ‰ ˆ ‰ 7 4 4 � 10. 5 , the sum of this geometric series is 4 � � � � á œ 5 5 5 5 4 16 64 1 � � ˆ ‰ 4 11. (5 1) , is the sum of two geometric series; the sum is � � � � � � � � á ˆ ‰ ˆ ‰ ˆ ‰ 5 5 5 3 4 9 8 7 # # " " " 10 5 3 23 1 1 � � " # # ˆ ‰ ˆ ‰ � œ � œ 3 12. (5 1) , is the difference of two geometric series; the sum is � � � � � � � � á ˆ ‰ ˆ ‰ ˆ ‰ 5 5 5 3 4 9 8 7 # # " " " 10 5 3 17 1 1 � � " # # ˆ ‰ ˆ ‰ � œ � œ 3 13. (1 1) , is the sum of two geometric series; the sum is � � � � � � � � á ˆ ‰ ˆ ‰ ˆ ‰ 1 1 1 5 4 25 8 1 5 # # " " " 2 1 5 17 1 1 6 6 � � " ˆ ‰ ˆ ‰ � œ � œ 5 14. 2 2 1 ; the sum of this geometric series is 2 � � � � á œ � � � � á œ 4 8 16 2 4 8 10 5 25 125 5 25 125 3 1 ˆ ‰ Š ‹ " � ˆ ‰ 2 5 15. Series is geometric with r 1 Converges to œ Ê � Ê œ 2 2 1 5 5 5 3 1 ¹ ¹ � 2 5 16. Series is geometric with r 3 3 1 Diverges œ � Ê � � Ê ¹ ¹ 17. Series is geometric with r 1 Converges to œ Ê � Ê œ 1 1 1 8 8 7 1 ¹ ¹ 1 8 1 8 � 18. Series is geometric with r 1 Converges to œ � Ê � � Ê œ � 2 2 2 3 3 5 1 ¹ ¹ � � � 2 3 2 3 ˆ ‰ 19. 0.23 20. 0.234 œ œ œ œ œ œ ! ! ˆ ‰ ˆ ‰ _ _ œ œ n 0 n 0 23 23 234 234 100 10 99 1000 10 999 n n 1 1 " " � � Š ‹ Š ‹ ˆ ‰ Š ‹ 23 234 100 1000 1000 100 21. 0.7 22. 0.d œ œ œ œ œ œ ! ! ˆ ‰ ˆ ‰ _ _ œ œ n 0 n 0 7 7 d d 10 10 9 10 10 9 n n 1 1 " " � � Š ‹ Š ‹ Š ‹ Š ‹ 7 d 10 10 10 10 23. 0.06 œ œ œ œ ! ˆ ‰ ˆ ‰ ˆ ‰ _ œ n 0 1 6 6 10 10 10 90 15 n 1 " " � Š ‹ Š ‹ 6 100 10 24. 1.414 1 1 1 œ � œ � œ � œ ! ˆ ‰ _ œ n 0 414 414 413 1000 10 999 999 n 1 " " � Š ‹ Š ‹ 414 1000 1000 Copyright © 2010 Pearson Education Inc Publishing as Addison-Wesley Section 10.2 Infinite Series 57 25. 1.24123 œ � œ � œ � œ � œ œ 124 123 124 124 123 124 123 100 10 10 100 100 10 10 100 99,900 99,900 33,300 n 1 123,999 41,333 ! ˆ ‰ _ œ n 0 " � � Š ‹ Š ‹ 123 10 10 26. 3.142857 3 3 3 œ � œ � œ � œ œ ! ˆ ‰ _ œ n 0 142,857 142,857 3,142,854 116,402 10 10 10 1 999,999 37,037 n 1 " � � Š ‹ Š ‹ 142,857 10 10 27. 1 0 diverges lim lim n n n 1 n 10 1 Ä_ Ä_ � œ œ Á Ê 28. 1 0 diverges lim lim lim lim n n n n n n 1 n 2 n 3 n 5n 6 2n 5 2 n n 2n 1 2 Ä_ Ä_ Ä_ Ä_ � � � � � � � � a b a ba b œ œ œ œ Á Ê 2 2 29. 0 test inconclusive lim n 1 n 4 Ä_ � œ Ê 30. 0 test inconclusive lim lim n n n 1 n 3 2n Ä_ Ä_ 2 � œ œ Ê 31. cos cos 0 1 0 diverges lim n 1 n Ä_ œ œ Á Ê 32. 1 0 diverges lim lim lim lim n n n n e e e 1 e n e 1 e 1 Ä_ Ä_ Ä_ Ä_ � � n n n n n n œ œ œ œ Á Ê 33. ln 0 diverges lim n 1 n Ä_ œ �_ Á Ê 34. cos n does not exist diverges lim nÄ_ 1 œ Ê 35. s 1 1 lim s k k 2 2 3 3 4 k 1 k k k 1 k 1 œ � � � � � � á � � � � œ � Ê ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ " " " " " " " " " " � � � k Ä _ lim 1 1, series converges to 1 œ � œ k Ä _ ˆ ‰ " � k 1 36. s 3 lim s k k 3 3 3 3 3 3 3 3 3 3 3 1 4 4 9 9 16 k k k 1 k 1 k 1 œ � � � � � � á � � � � œ � Ê ˆ ‰ ˆ ‰ ˆ ‰ Š ‹ Š ‹ a b a b a b � � � 2 2 2 2 2 k Ä _ lim 3 3, series converges to 3 œ � œ k Ä _ Š ‹ 3 ak 1 b � 2 Section 10.2 Infinite Series 579 25. 1.24123 œ � œ � œ � œ � œ œ 124 123 124 124 123 124 123 100 10 10 100 100 10 10 100 99,900 99,900 33,300 n 1 123,999 41,333 ! ˆ ‰ _ œ n 0 " � � Š ‹ Š ‹ 123 10 10 26. 3.142857 3 3 3 œ � œ � œ � œ œ ! ˆ ‰ _ œ n 0 142,857 142,857 3,142,854 116,402 10 10 10 1 999,999 37,037 n 1 " � � Š ‹ Š ‹ 142,857 10 10 27. 1 0 diverges lim lim n n n 1 n 10 1 Ä_ Ä_ � œ œ Á Ê 28. 1 0 diverges lim lim lim lim n n n n n n 1 n 2 n 3 n 5n 6 2n 5 2 n n 2n 1 2 Ä_ Ä_ Ä_ Ä_ � � � � � � � � a b a ba b œ œ œ œ Á Ê 2 2 29. 0 test inconclusive lim n 1 n 4 Ä_ � œ Ê 30. 0 test inconclusive lim lim n n n 1 n 3 2n Ä_ Ä_ 2 � œ œ Ê 31. cos cos 0 1 0 diverges lim n 1 n Ä_ œ œ Á Ê 32. 1 0 diverges lim lim lim lim n n n n e e e 1 e n e 1 e 1 Ä_ Ä_ Ä_ Ä_ � � n n n n n n œ œ œ œ Á Ê 33. ln 0 diverges lim n 1 n Ä_ œ �_ Á Ê 34. cos n does not exist diverges lim nÄ_ 1 œ Ê 35. s 1 1 lim s k k 2 2 3 3 4 k 1 k k k 1 k 1 œ � � � � � � á � � � � œ � Ê ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ " " " " " " " " " " � � � k Ä _ lim 1 1, series converges to 1 œ � œ k Ä _ ˆ ‰ " � k 1 36. s 3 lim s k k 3 3 3 3 3 3 3 3 3 3 3 1 4 4 9 9 16 k k k 1 k 1 k 1 œ � � � � � � á � � � � œ � Ê ˆ ‰ ˆ ‰ ˆ ‰ Š ‹ Š ‹ a b a b a b � � � 2 2 2 2 2 k Ä _ lim 3 3, series converges to 3 œ � œ k Ä _ Š ‹ 3 ak 1 b � 2 37. s ln 2 ln 1 ln 3 ln 2 ln 4 ln 3 ln k ln k 1 ln k 1 ln k k œ � � � � � � á � � � � � � Š ‹ Š ‹ Š ‹ Š ‹ Š ‹ È È È È È È È È È È ln k 1 ln 1 ln k 1 lim s lim ln k 1 ; series diverges œ � � œ � Ê œ � œ _ È È È È k k Ä _ Ä _ k 38. s tan 1 tan 0 tan 2 tan 1 tan 3 tan 2 tan k tan k 1 tan k 1 tan k k œ � � � � � � á � � � � � � a b a b a b a b a b a b a b tan k 1 tan 0 tan k 1 lim sk lim tan k 1 does not exist; series diverges œ � � œ � Ê œ � œ a b a b a b k k Ä _ Ä _ k _ lim 1 1, series converges to 1 œ � œ k Ä _ ˆ ‰ " � k 1 36. s 3 lim s k k 3 3 3 3 3 3 3 3 3 3 3 1 4 4 9 9 16 k k k 1 k 1 k 1 œ � � � � � � á � � � � œ � Ê ˆ ‰ ˆ ‰ ˆ ‰ Š ‹ Š ‹ a b a b a b � � � 2 2 2 2 2 k Ä _ lim 3 3, series converges to 3 œ � œ k Ä _ Š ‹ 3 ak 1 b � 2 37. s ln 2 ln 1 ln 3 ln 2 ln 4 ln 3 ln k ln k 1 ln k 1 ln k k œ � � � � � � á � � � � � � Š ‹ Š ‹ Š ‹ Š ‹ Š ‹ È È È È È È È È È È ln k 1 ln 1 ln k 1 lim s lim ln k 1 ; series diverges œ � � œ � Ê œ � œ _ È È È È k k Ä _ Ä _ k 38. s tan 1 tan 0 tan 2 tan 1 tan 3 tan 2 tan k tan k 1 tan k 1 tan k k œ � � � � � � á � � � � � � a b a b a b a b a b a b a b tan k 1 tan 0 tan k 1 lim sk lim tan k 1 does not exist; series diverges œ � � œ � Ê œ � œ a b a b a b k k Ä _ Ä _ 39. s cos cos cos cos cos cos k 1 1 1 1 1 1 1 1 1 1 1 1 2 3 3 4 4 5 œ � � � � � � á ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ � � � � � � cos cos cos cos cos � � � � œ � ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ � � � � � � � � # � # 1 1 1 1 1 1 1 1 1 1 k k 1 k 1 k 3 k 1 lim s lim cos , series converges to Ê œ � œ � œ k k Ä _ Ä _ k 3 k 3 2 6 6 1 1 ’ “ ˆ ‰ 1 1 1 1 1 � � # 40. s 5 4 6 5 7 6 k 3 k 2 k 4 k 3 k œ � � � � � � á � � � � � � � � Š ‹ Š ‹ Š ‹ Š ‹ Š ‹ È È È È È È È È È È k 4 2 lim s lim k 4 2 ; series diverges œ � � Ê œ � � œ _ È ’È “ k k Ä _ Ä _ k Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. lim 3 3, series converges to 3 œ � œ k Ä _ Š ‹ 3 ak 1 b � 2 37. s ln 2 ln 1 ln 3 ln 2 ln 4 ln 3 ln k ln k 1 ln k 1 ln k k œ � � � � � � á � � � � � � Š ‹ Š ‹ Š ‹ Š ‹ Š ‹ È È È È È È È È È È ln k 1 ln 1 ln k 1 lim s lim ln k 1 ; series diverges œ � � œ � Ê œ � œ _ È È È È k k Ä _ Ä _ k 38. s tan 1 tan 0 tan 2 tan 1 tan 3 tan 2 tan k tan k 1 tan k 1 tan k k œ � � � � � � á � � � � � � a b a b a b a b a b a b a b tan k 1 tan 0 tan k 1 lim sk lim tan k 1 does not exist; series diverges œ � � œ � Ê œ � œ a b a b a b k k Ä _ Ä _ 39. s cos cos cos cos cos cos k 1 1 1 1 1 1 1 1 1 1 1 1 2 3 3 4 4 5 œ � � � � � � á ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ � � � � � � cos cos cos cos cos � � � � œ � ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ � � � � � � � � # � # 1 1 1 1 1 1 1 1 1 1 k k 1 k 1 k 3 k 1 lim s lim cos , series converges to Ê œ � œ � œ k k Ä _ Ä _ k 3 k 3 2 6 6 1 1 ’ “ ˆ ‰ 1 1 1 1 1 � � # 40. s 5 4 6 5 7 6 k 3 k 2 k 4 k 3 k œ � � � � � � á � � � � � � � � Š ‹ Š ‹ Š ‹ Š ‹ Š ‹ È È È È È È È È È È k 4 2 lim s lim k 4 2 ; series diverges œ � � Ê œ � � œ _ È ’È “ k k Ä _ Ä _ k Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 43. 40n A B C D (2n 1) (2n 1) (2n 1) (2n 1) (2n 1) (2n 1) (2n 1) (2n 1) A(2n 1)(2n 1) B(2n 1) C(2n 1)(2n 1) D(2n 1) � � � � � � � � � � � � � � � � � œ � � � œ A(2n 1)(2n 1) B(2n 1) C(2n 1)(2n 1) D(2n 1) 40n Ê � � � � � � � � � œ # # # # A 8n 4n 2n 1 B 4n 4n 1 C 8n 4n 2n 1 D 4n 4n 1 40n Ê � � � � � � � � � � œ � � œ a b a b a b a b $ # # $ # # (8A 8C)n (4A 4B 4C 4D)n ( 2A 4B 2C 4D)n ( A B C D) 40n Ê � � � � � � � � � � � � � � � œ $ # 8A 8C 0 8A 8C 0 4A 4B 4C 4D 0 A B C D 0 2A 4B 2C 4D 40 A 2 A B C D 0 Ê Ê � œ � œ � � � œ � � � œ � � � � œ � � � � � � œ Ú Ú Ý Ý Ý Ý Û Û Ý Ý Ý Ý Ü Ü B C 2D 20 2B 2D 20 A B C D 0 4B 20 B 5 B D 0 � � œ � œ � � � � œ Ê Ê œ Ê œ � œ œ and D 5 C 0 and A 0. Hence, A C 0 A 5 C 5 0 œ � Ê Ê œ œ � œ � � � � œ œ ! ’ “ k n 1 œ 40n ( n 1) (2n 1) # � � 5 5 œ � œ � � � � � á � � � !’ “ Š ‹ k n 1 œ " " " " " " " " " " # � # � # # � � # � # � ( n 1) ( n 1) 1 9 9 5 5 (2(k 1) 1) ( k 1) ( k 1) 5 1 the sum is lim 5 1 5 œ � Ê � œ Š ‹ Š ‹ " " � � (2k 1) (2k 1) n Ä _ 44. s 1 2n 1 n (n 1) n (n 1) 4 4 9 9 16 (k 1) k k (k 1) k � " " " " " " " " " " " � � � � œ � Ê œ � � � � � � á � � � � ˆ ‰ ˆ ‰ ˆ ‰ ’ “ ’ “ lim s lim 1 1 Ê œ � œ k k Ä _ Ä _ k (k 1) ’ “ " � 45. s 1 1 k 2 2 4 3 3 k 1 k k k 1 k 1 œ � � � � � � á � � � � œ � Š ‹ Š ‹ Š ‹ Š ‹ Š ‹ " " " " " " " " " " � � � È È È È È È È È È È lim s lim 1 1 Ê œ � œ k k Ä _ Ä _ k k 1 Š ‹ " � È 46. s k œ � � � � � � á � � � � œ � ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ " " " " " " " " " " " " # # # # # # # # # # # # 1 k 1 1 k 1 k 1 k 1 1 k 1 Ê lim s œ � œ � k Ä _ k 1 " " " # # 47. sk ln 3 ln ln 4 ln 3 ln 5 ln 4 ln (k 1) ln k ln (k 2) ln (k 1) œ � � � � � � á � � � � ˆ ‰ ˆ ‰ ˆ ‰ Š ‹ Š ‹ " " " " " " " " " " # � � � lim s œ � � Ê œ � " " " # � # ln ln (k 2) ln k k Ä _ 48. s tan (1) tan (2) tan (2) tan (3) tan (k 1) tan (k) k œ � � � � á � � � c d c d c d �" �" �" �" �" �" tan (k) tan (k 1) tan (1) tan (k 1) lim s tan (1) � � � œ � � Ê œ � œ � œ � c d �" �" �" �" �" # # k Ä _ k 4 4 1 1 1 1 49. convergent geometric series with sum 2 2 " � � 1 2 2 1 Š ‹ È È 2 œ œ � È 50. divergent geometric series with r 2 1 51. convergent geometric series with sum 1 k k œ È � œ Š ‹ Š ‹ 3 1 � � Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. A 5 C 5 0 � � œ œ ’ “ n 1 œ ( ) ( ) � 5 5 œ � œ � � � � � á � � � !’ “ Š ‹ k n 1 œ " " " " " " " " " " # � # � # # � � # � # � ( n 1) ( n 1) 1 9 9 5 5 (2(k 1) 1) ( k 1) ( k 1) 5 1 the sum is lim 5 1 5 œ � Ê � œ Š ‹ Š ‹ " " � � (2k 1) (2k 1) n Ä _ 44. s 1 2n 1 n (n 1) n (n 1) 4 4 9 9 16 (k 1) k k (k 1) k � " " " " " " " " " " " � � � � œ � Ê œ � � � � � � á � � � � ˆ ‰ ˆ ‰ ˆ ‰ ’ “ ’ “ lim s lim 1 1 Ê œ � œ k k Ä _ Ä _ k (k 1) ’ “ " � 45. s 1 1 k 2 2 4 3 3 k 1 k k k 1 k 1 œ � � � � � � á � � � � œ � Š ‹ Š ‹ Š ‹ Š ‹ Š ‹ " " " " " " " " " " � � � È È È È È È È È È È lim s lim 1 1 Ê œ � œ k k Ä _ Ä _ k k 1 Š ‹ " � È 46. s k œ � � � � � � á � � � � œ � ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ " " " " " " " " " " " " # # # # # # # # # # # # 1 k 1 1 k 1 k 1 k 1 1 k 1 Ê lim s œ � œ � k Ä _ k 1 " " " # # 47. sk ln 3 ln ln 4 ln 3 ln 5 ln 4 ln (k 1) ln k ln (k 2) ln (k 1) œ � � � � � � á � � � � ˆ ‰ ˆ ‰ ˆ ‰ Š ‹ Š ‹ " " " " " " " " " " # � � � lim s œ � � Ê œ � " " " # � # ln ln (k 2) ln k k Ä _ 48. s tan (1) tan (2) tan (2) tan (3) tan (k 1) tan (k) k œ � � � � á � � � c d c d c d �" �" �" �" �" �" tan (k) tan (k 1) tan (1) tan (k 1) lim s tan (1) � � � œ � � Ê œ � œ � œ � c d �" �" �" �" �" # # k Ä _ k 4 4 1 1 1 1 49. convergent geometric series with sum 2 2 " � � 1 2 2 1 Š ‹ È È 2 œ œ � È 50. divergent geometric series with r 2 1 51. convergent geometric series with sum 1 k k œ È � œ Š ‹ Š ‹ 3 1 � � Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. A B C D 0 � � � � œ Ý Ý Ü Ü A B C D 0 � � � � œ and D 5 C 0 and A 0. Hence, A C 0 A 5 C 5 0 œ � Ê Ê œ œ � œ � � � � œ œ ! ’ “ k n 1 œ 40n ( n 1) (2n 1) # � � 5 5 œ � œ � � � � � á � � � !’ “ Š ‹ k n 1 œ " " " " " " " " " " # � # � # # � � # � # � ( n 1) ( n 1) 1 9 9 5 5 (2(k 1) 1) ( k 1) ( k 1) 5 1 the sum is lim 5 1 5 œ � Ê � œ Š ‹ Š ‹ " " � � (2k 1) (2k 1) n Ä _ 44. s 1 2n 1 n (n 1) n (n 1) 4 4 9 9 16 (k 1) k k (k 1) k � " " " " " " " " " " " � � � � œ � Ê œ � � � � � � á � � � � ˆ ‰ ˆ ‰ ˆ ‰ ’ “ ’ “ lim s lim 1 1 Ê œ � œ k k Ä _ Ä _ k (k 1) ’ “ " � 45. s 1 1 k 2 2 4 3 3 k 1 k k k 1 k 1 œ � � � � � � á � � � � œ � Š ‹ Š ‹ Š ‹ Š ‹ Š ‹ " " " " " " " " " " � � � È È È È È È È È È È lim s lim 1 1 Ê œ � œ k k Ä _ Ä _ k k 1 Š ‹ " � È 46. s k œ � � � � � � á � � � � œ � ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ " " " " " " " " " " " " # # # # # # # # # # # # 1 k 1 1 k 1 k 1 k 1 1 k 1 Ê lim s œ � œ � k Ä _ k 1 " " " # # 47. sk ln 3 ln ln 4 ln 3 ln 5 ln 4 ln (k 1) ln k ln (k 2) ln (k 1) œ � � � � � � á � � � � ˆ ‰ ˆ ‰ ˆ ‰ Š ‹ Š ‹ " " " " " " " " " " # � � � lim s œ � � Ê œ � " " " # � # ln ln (k 2) ln k k Ä _ 48. s tan (1) tan (2) tan (2) tan (3) tan (k 1) tan (k) k œ � � � � á � � � c d c d c d �" �" �" �" �" �" tan (k) tan (k 1) tan (1) tan (k 1) lim s tan (1) � � � œ � � Ê œ � œ � œ � c d �" �" �" �" �" # # k Ä _ k 4 4 1 1 1 1 49. convergent geometric series with sum 2 2 " � � 1 2 2 1 Š ‹ È È 2 œ œ � È 50. divergent geometric series with r 2 1 51. convergent geometric series with sum 1 k k œ È � œ Š ‹ Š ‹ 3 1 � � Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.2 Infinite Series 581 52. lim ( 1) n 0 diverges 53. lim cos (n ) lim ( 1) 0 diverges n n n Ä _ Ä _ Ä _ � Á Ê œ � Á Ê n 1 n � 1 54. cos (n ) ( 1) convergent geometric series with sum 1 œ � Ê œ n 1 5 6 " � Š ‹ � 5 55. convergent geometric series with sum " � � 1 e e 1 Š ‹ e œ 56. lim ln 0 diverges n Ä _ " 3n œ �_ Á Ê 57. convergent geometric series with sum 2 2 20 18 2 1 9 9 9 � Š ‹ 10 � œ � œ 58. convergent geometric series with sum " � � 1 x x 1 Š ‹ x œ 59. difference of two geometric series with sum 3 " " � � # # 1 1 3 3 Š ‹ Š ‹ 2 3 3 � œ � œ 60. lim 1 lim 1 e 0 diverges n n Ä _ Ä _ ˆ ‰ ˆ ‰ � œ � œ Á Ê " �" �" n n n n 61. lim 0 diverges 62. lim lim lim n diverges n n n n Ä _ Ä _ Ä _ Ä _ n! n n n n 1000 n! 1 n n n œ _ Á Ê œ � œ _ Ê † † â #â 63. ; both and are geometric series, and both conve ! ! ! ! ! ! ! ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ _ _ _ _ _ _ _ œ œ œ œ œ œ œ n 1 n 1 n 1 n 1 n 1 n 1 n 1 2 3 2 3 1 3 1 3 4 4 4 2 4 2 4 n n n n n n n n n n n � œ � œ � œ rge since r 1 and r 1, respectivley 1 and 3 œ Ê � œ Ê � Ê œ œ œ œ Ê 1 1 3 3 1 3 2 2 4 4 2 4 n n 1 1 ¹ ¹ ¹ ¹ ! ! ˆ ‰ ˆ ‰ _ _ œ œ n 1 n 1 1 3 2 4 1 3 2 4 � � 1 3 4 by Theorem 8, part (1) ! _ œ n 1 2 3 4 n n n� œ �œ 64. 1 0 diverges by n term test for divergence lim lim lim n n n 2 4 1 3 4 1 th Ä_ Ä_ Ä_ � � � " � " � " � " n n n n 2 1 n 4 2 n 3 3 n n 4 4 n n œ œ œ œ Á Ê ˆ ‰ ˆ ‰ 65. ln ln (n) ln (n 1) s ln (1) ln (2) ln (2) ln (3) ln (3) ln (4) ! ! ˆ ‰ c d c d c d c d _ _ œ œ n 1 n 1 n n 1 � œ � � Ê œ � � � � � � á k ln (k 1) ln (k) ln (k) ln (k 1) ln (k 1) lim s , diverges � � � � � � œ � � Ê œ �_ Ê c d c d k Ä _ k 66. lim a lim ln ln 0 diverges n n Ä _ Ä _ n n 2n 1 œ œ Á Ê ˆ ‰ ˆ ‰ � # " 67. convergent geometric series with sum " � � 1 e ˆ ‰ e œ 1 1 Section 10.3: 63. ; both and are geometric series, and both conve ! ! ! ! ! ! ! ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ œ œ œ œ œ œ œ n 1 n 1 n 1 n 1 n 1 n 1 n 1 4 4 4 2 4 2 4 n n n œ � œ � œ rge since r 1 and r 1, respectivley 1 and 3 œ Ê � œ Ê � Ê œ œ œ œ Ê 1 1 3 3 1 3 2 2 4 4 2 4 n n 1 1 ¹ ¹ ¹ ¹ ! ! ˆ ‰ ˆ ‰ _ _ œ œ n 1 n 1 1 3 2 4 1 3 2 4 � � 1 3 4 by Theorem 8, part (1) ! _ œ n 1 2 3 4 n n n� œ �œ 64. 1 0 diverges by n term test for divergence lim lim lim n n n 2 4 1 3 4 1 th Ä_ Ä_ Ä_ � � � " � " � " � " n n n n 2 1 n 4 2 n 3 3 n n 4 4 n n œ œ œ œ Á Ê ˆ ‰ ˆ ‰ 65. ln ln (n) ln (n 1) s ln (1) ln (2) ln (2) ln (3) ln (3) ln (4) ! ! ˆ ‰ c d c d c d c d _ _ œ œ n 1 n 1 n n 1 � œ � � Ê œ � � � � � � á k ln (k 1) ln (k) ln (k) ln (k 1) ln (k 1) lim s , diverges � � � � � � œ � � Ê œ �_ Ê c d c d k Ä _ k 66. lim a lim ln ln 0 diverges n n Ä _ Ä _ n n 2n 1 œ œ Á Ê ˆ ‰ ˆ ‰ � # " 67. convergent geometric series with sum " � � 1 e ˆ ‰ e œ 1 1 68. divergent geometric series with r 1 k k œ ¸ � e 23.141 22.459 1e 69. ( 1) x ( x) ; a 1, r x; converges to for x 1 ! ! k k _ _ œ œ n 0 n 0 � œ � œ œ � œ � n n n " " � � � 1 ( x) 1 x 70. ( 1) x x ; a 1, r x ; converges to for x 1 ! ! a b k k _ _ œ œ n 0 n 0 � œ � œ œ � � n 2n n 1 x # # " � Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 582 Chapter 10 Infinite Sequences and Series 71. a 3, r ; converges to for 1 1 or 1 x 3 œ œ œ � � � � � � x 1 3 6 x 1 3 x � �" # � # � Š ‹ x 72. ; a , r ; converges to ! ! ˆ ‰ ˆ ‰ _ _ œ œ n 0 n 0 ( 1) 3 sin x 3 sin x 3 sin x n n 1 � # � # � # � " " �" " �" � n œ œ œ ˆ ‰ Š ‹ 3 sin x for all x since for all x œ œ Ÿ Ÿ 3 sin x 3 sin x 2(4 sin x) 8 2 sin x 4 3 sin x � � " " " � � � # ˆ ‰ 73. a 1, r 2x; converges to for 2x 1 or x œ œ � � " " � # 1 2x k k k k 74. a 1, r ; converges to for 1 or x 1. œ œ � œ � � " " � � x x 1 x 1 x 1 Š ‹ x ¸ ¸ k k 75. a 1, r (x 1) ; converges to for x 1 1 or 2 x 0 œ œ � � œ � � � � � n 1 (x 1) x " " � � # � k k 76. a 1, r ; converges to for 1 or 1 x 5 œ œ œ � � � 3 x 2 3 x 1 x 1 � " � # � # � Š 3 x ‹ ¸ ¸ 77. a 1, r sin x; converges to for x (2k 1) , k an integer œ œ Á � " � # 1 sin x 1 78. a 1, r ln x; converges to for ln x 1 or e x e œ œ � � � " � �" 1 ln x k k 79. (a) (b) (c) ! ! ! _ _ _ œ� œ œ n 2 n 0 n 5 " " " � � � � � � # (n 4)(n 5) (n 2)(n 3) (n 3)(n ) 80. (a) (b) (c) ! ! ! _ _ _ œ� œ œ n 1 n 3 n 20 5 5 5 (n 2)(n 3) (n 2)(n 1) (n 19)(n 18) � � � � � � 81. (a) one example is 1 " " " " # � � � � � á œ œ 4 8 16 1 Š ‹ Š ‹ (b) one example is 3 � � � � � á œ œ � 3 3 3 3 4 8 16 1 # � � Š ‹ Š ‹ 3 (c) one example is 1 1 0. � � � � � á œ � œ " " " " # � 4 8 16 1 Š ‹ Š ‹ 82. The series k is a geometric series whose sum is k where k can be any positive or negative number ! ˆ ‰ _ œ n 0 1 2 n 1 1 � � Š ‹ Š ‹ k œ 83. Let a b . Then a b 1, while (1) diverges. n n n n n n a b œ œ œ œ œ œ ˆ ‰ ˆ ‰ ! ! ! ! ! Š ‹ " " # # _ _ _ _ _ œ œ œ œ œ n 1 n 1 n 1 n 1 n 1 n n 584 Chapter 10 Infinite Sequences and Series 3. f x is positive, continuous, and decreasing for x 1; dx lim dx lim tan a b ’ “ œ œ œ 1 1 1 1 x x 4 x 4 x 4 2 2 1 2 2 2 � � � � ' ' 1 1 b 1 b b b Ä _ Ä _ lim tan tan tan dx converges converges œ � œ � Ê Ê b Ä _ ˆ ‰ ! 1 b 1 1 1 1 1 1 2 2 2 2 4 2 2 x 4 n 4 1 1 1 n 1 � � � � � œ _ 1 ' 1 2 2 4. f x is positive, continuous, and decreasing for x 1; dx lim dx lim ln x 4 a b ’ “ œ œ œ l � l 1 1 1 x 4 x 4 x 4 � � � ' ' 1 1 b 1 b b b Ä _ Ä _ lim ln b 4 ln 5 dx diverges diverges œ l � l � œ _ Ê Ê b Ä _ a b ! ' 1 1 1 x 4 n 4 n 1 � � œ _ 5. f x e is positive, continuous, and decreasing for x 1; e dx lim e dx lim e a b ’ “ œ œ œ � � � � � 2x 2x 2x 2x 1 2 ' ' 1 1 b 1 b b b Ä _ Ä _ lim e dx converges e converges œ � � œ Ê Ê b Ä _ ˆ ‰ ! 1 1 1 2e 2e 2e 2x 2n n 1 2b 2 2 ' 1 � � œ _ 6. f x is positive, continuous, and decreasing for x 2; dx lim dx lim a b ’ “ œ œ œ � 1 1 1 1 x ln x x ln x x ln x ln x a b a b a b 2 2 2 ' ' 2 2 b 2 b b b Ä _ Ä _ lim dx converges converges œ � � œ Ê Ê b Ä _ ˆ ‰ ! 1 1 1 1 1 ln b ln 2 ln 2 x ln x n 2 n ln n ' 2 a b a b 2 2 œ _ 7. f x is positive and continuous for x 1, f x 0 for x 2, thus f is decreasing for x 3; a b a b œ œ � � x 4 x x 4 x 4 2 2 2 2 � w � � a b dx lim dx lim ln x 4 lim ln b 4 ln 13 dx ' ' ' 3 3 3 b 3 b x x 1 1 1 x x 4 x 4 2 2 2 x 4 2 2 2 2 2 � � � œ œ � œ � � œ _ Ê b b b Ä _ Ä _ Ä _ ’ “ a b a b a b ˆ ‰ diverges diverges diverges Ê Ê œ � � ! ! ! n 3 n 1 n 3 n n 1 2 n n 4 n 4 5 8 n 4 œ œ œ _ _ _ � � � 2 2 2 8. f x is positive and continuous for x 2, f x 0 for x e, thus f is decreasing for x 3; a b a b œ œ � � ln x 2 ln x x x 2 2 2 w � dx lim dx lim 2 ln x lim 2 ln b 2 ln 3 dx ' ' ' 3 3 3 b 3 b ln x ln x ln x x x x 2 2 2 œ œ œ � œ _ Ê b b b Ä _ Ä _ Ä _ ’ “ a b a b a b a b diverges diverges diverges Ê Ê œ � ! ! ! n 3 n 2 n 3 ln n ln n ln 4 ln n n n 2 n œ œ œ _ _ _ 2 2 2 9. f x is positive and continuous for x 1, f x 0 for x 6, thus f is decreasing for x 7; a b a b œ œ � � x e 3e x x 6 2 x 3 x 3 w � a � b dx lim dx lim lim ' ' 7 7 b 7 b x x 3x 18x 54 3b 18b 54 327 e e e e e e e 2 2 2 2 x 3 x 3 x 3 x 3 x 3 b 3 7 3 œ œ � � � œ � œ b b b Ä _ Ä _ Ä _