ISMT12_C10_A

CHAPTER 10 INFINITE SEQUENCES AND SERIES10.1 SEQUENCES

1. a " œ 1 c 1 1 # œ 0, a # œ "c 2 #

# œc " 4 , a $ œ 1 c 3 3 # œc 2 9 , a

%

œ 1 c

4 4 #

œc 16 3

2. a " œ 1 1! œ 1, a # œ " #

! œ "

2 , a $ œ 1 3! œ 1 6 , a

%

œ 1 4! œ 1 24

3. a " œ ( c 1) # #c 1 œ 1, a # œ ( c" ) $ 4 c 1 œc " 3 , a $ œ ( c 1) % 6 c 1 œ " 5 , a

%

œ ( c

1)

& 8 c

1 œc

" 7

4. a " œ 2 b ( c 1) " œ 1, a # œ 2 b ( c 1) # œ 3, a $ œ 2 b ( c 1) $ œ 1, a %

œ 2 b ( c 1) % œ

3

5. a " œ 2 # # œ " # , a # œ 2 # 2 $ œ " # , a $ œ 2 $ # %

œ " # , a

%

œ 2

% 2 &

œ " #

6. a " œ 2 c" # œ " #

, a # œ 2 # c 1 2 # œ 3 4 , a $ œ 2 $ c 1 2 $ œ 7 8 , a

%

œ 2 %

c" 2

%

œ 15 16 7. a " œ 1, a # œ 1 b " # œ 3 # , a $ œ 3 # b " # # œ 7 4 , a % œ 7 4 b " # $ œ 15 8 , a & œ 15 8 b " #

%

œ 31 16 , a '

œ 63

32 , a ( œ 127 64 , a ) œ 255

128 , a * œ 511 256 , a

"!

œ 1023

512

8. a " œ 1, a # œ " # , a $ œ ˆ 3 # " ‰ œ " 6 , a % œ ˆ 4 " 6 ‰ œ " # 4 , a & œ ˆ # 5 " 4

‰ œ " 1 # 0 , a ' œ " 7 #

0 , a ( œ " 5040 , a )

œ

" 40,320

, a * œ " 362,880 , a

"!

œ

" 3,628,800

9. a " œ 2, a # œ ( c 1) # (2) # œ 1, a $ œ ( c

1) $ (1)

2 œc " # , a % œ ( c 1) % ˆ # ‰ , a &

& ˆ " 4

‰

, a ' , a ( , a ) , a * , a

"!

c " #

œc " 4 œ ( c 1) #

c

œ

" 8 œ " 16 œc 3 " # œc 64 " œ " 1 #

8 œ

" 256

10. a " œc 2, a # œ 1( † c 2) # œc 1, a $ œ 2( † c

1) 3 œc 2 3 , a % œ 3 † ˆ c 4 2 3

‰ œc " #

, a & œ 4 † ˆ c

5 " #

‰ œc 2

5 , a '

œc

" 3

, a ( œc 2 7 , a )

œc " 4 , a * œc 2

9 , a

"!

œc " 5

11. a " œ 1, a # œ 1, a $ œ 1 b 1 œ 2, a % œ 2 b 1 œ 3, a & œ 3 b 2 œ 5, a ' œ 8, a ( œ 13, a ) œ 21, a * œ 34, a "!

œ

55

12. a " œ 2, a # œc 1, a $ œc " # , a % œ ˆ c

c 1

" # ‰ œ " # , a & œ ˆ c ˆ " #

‰ " #

‰

œc 1, a ' œc 2, a ( œ 2, a ) œc 1, a * œc " # , a

"!

œ

" #

13. a n œ ( c 1) n b

1 , n œ 1, 2, á 14. a n

œ ( c 1) n , n œ 1, 2, á 15. a n œ ( c 1) n b 1 n # , n œ 1, 2, á 16. a n

œ ( c" )

n 1b n

#

, n œ 1, 2, á 17. a n œ 3n a 2 n b 1c

2 b , n œ 1, 2, á 18. a n

œ 2n nn a c

5 b

1

b , n œ 1, 2,

á

19. a n œ n #

c 1, n œ 1, 2, á 20. a n

œ n c 4, n œ 1, 2, á 21. a n œ 4n c 3, n œ 1, 2, á 22. a n

œ 4n c 2, n œ 1, 2,

á

23. a n œ 3n n! b 2 , n œ 1, 2, á 24. a n

œ n

3 5

n 1b

, n œ 1, 2, á Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

570 Chapter 10 Infinite Sequences and Series25. a n œ 1 b ( c

# 1) n b

1 , n œ 1, 2, á 26. a n

œ n c " # b ( # c 1)

n ˆ " #

‰

œÚ n #

Û , n œ 1, 2,

á

27. n lim Ä _

2 b (0.1) n

œ 2 Ê converges (Theorem 5, #4)

28. n lim Ä_ n b ( c" ) n n œ n lim Ä_

1 b ( c 1)

n n

œ 1 Ê

converges

29. n lim Ä_ "c 2n 1 b# n

œ n lim Ä_ ˆ ˆ " n " n

‰ ‰

n lim Ä_

1 converges c b

2 2

œ c

2 #

œ c Ê

30. n lim Ä_ 2n

b" 1 c

3 È

n

œ n lim Ä_

2 È n

b

Š È "

n

‹

Š È "

n

c

3

‹

œ c_ Ê

diverges

31. n lim Ä_ "c

5n

% n % b

8n

$

œ n lim Ä_

Š n " % ‹

c

5

1

b

ˆ 8 n

‰

œ c 5 Ê

converges

32. n lim Ä_ n b 3 n #

b 5n b 6 œ n lim Ä_ n b 3

(n b 3)(n b 2) œ n lim Ä_

" n

b# œ 0 Ê

converges

33. n lim Ä_ n # c 2n b

1 n c 1 œ n lim Ä_ (n c 1)(n n c

c 1) 1

œ n lim Ä_

(n c 1) œ_ Ê

diverges

34 n lim Ä_ "c

n

$ 70 c

4n

#

œ n lim Ä_

Š n

"

#

‹

c

n

Š 70 n

#

‹

c

4

œ_ Ê

diverges

35. n lim Ä_ a 1 b ( c 1) n b does not exist Ê diverges 36. n lim Ä_

( c 1) n

ˆ 1 c " n

‰ does not exist Ê diverges

37. n lim Ä_ ˆ n

# b" n ‰ˆ 1 c " n ‰ œ n lim Ä_

ˆ " # b # " n ‰ˆ 1 c " n

‰ œ " #

Ê

converges

38. n lim Ä_ ˆ 2 c # " n ‰ˆ 3 b # " n

‰ œ 6 Ê converges 39. n lim Ä_

( c"

)

n b 1 # n c 1

œ 0 Ê converges

40. n lim Ä_ ˆ c "

# ‰ n œ n lim Ä_

( c" )n #

n

œ 0 Ê converges

41. n lim Ä_ É n 2n b 1 œ É n lim Ä_ 2n n b

1 œ Ê n lim Ä_

Š 1 b 2

" n

‹

œ È 2 Ê

converges

42. n lim Ä_ " (0.9) n

œ n lim Ä_

ˆ " 9

0

‰n

œ_ Ê diverges

43. n lim Ä_ sin ˆ 1 # b " n ‰ œ sin Š n lim Ä_

ˆ 1 # b " n

‰ ‹ œ sin 1 #

œ 1 Ê

converges

44. n lim Ä_ n 1 cos(n 1 ) œ n lim Ä_

(n 1 )( c 1) n

does not exist Ê diverges

45. n lim Ä _

sin n n œ 0 because c " n Ÿ sin n

n Ÿ " n

Ê converges by the Sandwich Theorem for sequences

46. n lim Ä _

sin # n # n œ 0 because 0 Ÿ sin #

n # n Ÿ " #

n

Ê

converges by the Sandwich Theorem for sequences

47. n lim Ä_ n

# n œ n lim Ä_

" #

n

ln 2 œ 0 Ê

converges (using l'Hopital's ^

rule)

Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

Section 10.1 Sequences 57148. n lim Ä_ 3 n n $ œ n lim Ä_ 3 n ln 3

3n #

œ n lim Ä_ 3 n (ln 3) # 6n œ n lim Ä_

3 n

(ln 3) 6

diverges (using l'Hopital's ^

rule) $

œ_ Ê

49. n lim Ä_ ln (n b"

È

n

)

œ n lim Ä_ ˆ n b "

1

‰

Š #

È "

n

‹

œ n lim Ä_ 2 È

n n b

1

œ n lim Ä_

Š È 2

n

‹

1

b

Š n ‹

œ 0 Ê

converges

50. n lim Ä_ n lim Ä_

1 converges

"

ln n ln 2n

œ ˆ ˆ 2n " n 2

‰ ‰

œ Ê

51. n lim Ä _

8 1 nÎ œ 1 Ê

converges (Theorem 5, #3)

52. n lim Ä _

(0.03) 1 nÎ œ 1 Ê


53. n lim Ä _

ˆ 1 b 7 n

‰ n œ e (

Ê converges (Theorem 5, #5)

54. n lim Ä_ ˆ 1 c " n ‰ n

œ n lim Ä_

’ 1 b ( c" n

) “ n

œ e c"

Ê converges (Theorem 5, #5)

55. n lim Ä_ È

n 10n œ n lim Ä_

10 1 Î n † n 1 Î n œ 1 †

1 œ 1 Ê converges (Theorem 5, #3 and #2)

56. n lim Ä_ È n

n #

œ n lim Ä_

ˆ ‰È n

n #

œ 1 # œ 1 Ê


57. n lim Ä _

ˆ 3

n ‰ 1 Î

n œ n

lim Ä_

3

1 Î

n

n

lim Ä_

n

1 Î

n

œ " 1

œ 1 Ê converges (Theorem 5, #3 and #2)

58. n lim Ä_ (n b 4) 1 ÎÐ n b 4 Ñ œ x lim Ä_

x 1 Î

x œ 1 Ê converges; (let x œ n b 4, then use Theorem 5, #2)

59. n lim Ä _

ln n n 1 Î n œ n

lim Ä_

ln n

n

lim Ä_

n

1 Î

n œ _

1 œ_ Ê diverges (Theorem 5, #2)

60. n lim Ä_ c ln n c ln (n b 1) d œ n lim Ä_ ln ˆ n b n 1 ‰ œ ln Š n lim Ä_

n b n

1 ‹ œ ln 1 œ 0 Ê converges

61. n lim Ä_ È n 4 n n œ n lim Ä_

4 È

n

n œ 4 †

1 œ 4 Ê converges (Theorem 5, #2)

62. n lim Ä_ È

n 3 2n b 1 œ n lim Ä_ 3 2 b a 1 Î n b œ n lim Ä_

3 # † 3 1 Î n œ 9 †

1 œ 9 Ê converges (Theorem 5, #3)

63. n lim Ä_ n! n n œ n lim Ä_ " † 2 †

3 â (n c

1)(n) nnn † † â

nn †

Ÿ n lim Ä_ ˆ " n ‰

œ 0 and n! n n 0 Ê n lim Ä_

n! n

n

œ 0 Ê

converges

64. n lim Ä _

( c 4)

n n!

œ 0 Ê


65. n lim Ä_ n!

106n

œ n lim Ä_

" Š (10 n! '

)

n

‹

œ_ Ê diverges (Theorem 5, #6)

66. n lim Ä_ n!

2 3n n œ n lim Ä_

ˆ 6

n! " n

‰

œ_ Ê diverges (Theorem 5, #6)

67. n lim Ä_ ˆ " n ‰ 1 ÎÐ lnn

Ñ

œ n lim Ä_ exp ˆ ln " n ln ˆ " n ‰ ‰ œ n lim Ä_

exp ˆ ln 1 ln c

n

ln n ‰ œ e c"

Ê

converges


572 Chapter 10 Infinite Sequences and Series68. n lim Ä_ ln ˆ 1 b " n ‰ n œ ln Š n lim Ä_

ˆ 1 b " n

‰ n

‹ œ ln e œ 1 Ê converges (Theorem 5, #5)

69. n lim Ä_ ˆ 3n 3n b" c 1 ‰ n œ n lim Ä_ exp ˆ n ln ˆ 3n

3n b" c

1

‰ ‰ œ

n lim Ä_

exp

Š ln (3n b 1) c "

ln (3n c 1) ‹ n

œ n lim Ä_ exp : 3n 3 b cŠ 1 c n

"

# 3n ‹ 3 c

1

;

œ n lim Ä_

exp Š (3n b 1)(3n 6n #

c

1) ‹ œ exp ˆ 6 9

‰

œ e #Î$

Ê converges 70. n lim Ä_ ˆ n b n 1 ‰ n œ n lim Ä_ exp ˆ n ln ‰ˆ n b

n

1

‰ œ n lim Ä_ exp Š ln n c ˆ ln ‰

(n b

1)

‹

œ

n lim Ä_

exp

: " n Š c c

" b "

#

œ n lim Ä _

exp c œ e Ê

converges " n

n n

‹ 1

;

Š n

# n(n b

1)

‹ c"

71. n lim Ä_ ˆ 2n x

b n

1 ‰ 1 Î

n œ n lim Ä_ x ˆ # n " b 1 ‰ 1 Î

n

œ x n lim Ä_ exp ˆ " n ln ˆ # n "

b

1 ‰ ‰ œ

x n lim Ä_

exp

Š c ln (2n n

b 1) ‹ œ x n lim Ä _

exp ˆ 2n c

b 2

1

‰ œ xe ! œ x, x 0 Ê converges 72. n lim Ä_ ˆ 1 c n " # ‰ n

œ n lim Ä_ exp ˆ n ln ˆ 1 c n " #

‰

‰ œ n lim Ä_ exp : ln Š 1 ˆ " n c ‰

n

"

#

‹ ; œ n lim Ä_

exp

– Š n 2

$ ‹ Š ‚ c

Š n 1 " #

" n

œ n lim Ä _

exp œ e œ 1 Ê

converges c ‹

#

‹

—

ˆ n c

#

c 2n 1 ‰ ! 73. n lim Ä_ 3 n †

6 n 2 c n

† n! œ n lim Ä_

36

n n!

œ 0 Ê


74. n lim Ä_ ˆ ˆ 10 9 ‰ n 10 11 b ‰ ˆ n 11 12 ‰ n œ n lim Ä_ ˆ 12 11 ‰ n ˆ ˆ 10 9 12 11 ‰ ‰ n n b ˆ ˆ 10 11 12 11 ‰ n ‰ n ˆ 11 12 ‰ n œ n lim Ä_

ˆ 108 110

ˆ 120 121 ‰

n

‰

b n

1

œ 0 Ê

converg

es

(Theorem 5, #4)

75. n lim Ä_ tanh n œ n lim Ä_ e n c e c n e n b e c

n œ n lim Ä_ e 2n c"

e 2n b

1 œ n lim Ä_ 2e

2n 2e

2n

œ n lim Ä_

" œ 1 Ê converges

76. n lim Ä_ sinh (ln n) œ n lim Ä_ e ln n c

e

c

ln n 2

œ n lim Ä_

n c

#

"ˆ n

‰

œ_ Ê diverges

77. n lim Ä_ n # 2n sin

c "ˆ 1 n

‰

œ n lim Ä_ sin ˆ " n ‰ Š 2 n c n "

# ‹ œ n lim Ä_ c

ˆ cos ‰ˆ " n

‰ Š n "

#

‹

Š c n 2 # b

n

2

$

‹

œ n lim Ä_

c c# cos b

ˆ ˆ " n 2 n

‰ ‰

œ " #

Ê

converges

78. n lim Ä_ n ˆ 1 c cos " n ‰ œ n lim Ä_ ˆ "ccos

ˆ " n

‰

n lim Ä_ " n

‰

œ < sin ‘ˆ " n ‰ Š n "

#

‹

Š n

"

#

‹

œ n lim Ä_

sin ˆ " n

‰ œ 0 Ê converges

79. n lim Ä_ È n sin Š È 1 n ‹ œ n lim Ä_ sin Š È 1 n ‹ 1 È

n

œ n lim Ä_ cos

Š È

1 n ‹Š c 2n

1

c

2n

3 1

Î

2

‹

3 Î

2

œ n lim Ä_

cos cos 0 1 converges

Š È 1

n

‹ œ œ Ê

80. n lim Ä_ a 3 n b 5 n b 1n Î œ n lim Ä_ exp ’ ln a 3 n b 5 n b 1n Î “ œ n lim Ä_ exp ’ ln3 a n n b 5 n

b “

œ

n lim Ä_

exp

– 3 n ln3 3 b

5 n

ln5 b 1

5

—

œ n lim Ä_ exp œ n lim Ä_

exp œ exp ln 5 œ

5 n n ’ Š 3 5

n n ˆ ‹

3 5 ln n n

‰ 3 b b

1 ln 5

“ ’ ˆ 3 5

‰ ˆ n 3 5 ln ‰

n 3 b b

1

ln 5

“ a b

81. n lim Ä_ tan c" n œ 1 # Ê converges 82. n lim Ä_

" Èn

tan c"

n œ 0 †

1 #

œ 0 Ê converges


Section 10.1 Sequences 57383. n lim Ä_ ˆ " 3 ‰ n b " È 2 n

œ n lim Ä_

Š ˆ " 3

‰ n

b Š È "

2

‹ n

‹ œ 0 Ê


84. n lim Ä_ È

n n #

b n œ n lim Ä_ exp ’ ln a n #

n b n

b “ œ n lim Ä_

exp ˆ 2n n #

b b

n 1 ‰ œ e ! œ 1 Ê

converges

85. n lim Ä_ (ln n) #!! n œ n lim Ä_ 200 (ln n) "** n œ n lim Ä_ 200 †

199 (ln n)

"*) n œá œ n lim Ä_

200! n

œ 0 Ê

converges

86. n lim Ä_ (ln n) & È n œ n lim Ä_ – Š Š 5(ln #È n

"

n)

n

%

‹

‹

—

œ n lim Ä_ 10(ln n) % È n œ n lim Ä_ 80(ln n)

$ È n œá œ n lim Ä_

3840

È

n

œ 0 Ê

converges

87. n lim Ä_ Š n c È n # c n ‹ œ n lim Ä_ Š n c È

n #

c n ‹Š n n b È

n #

c b È n # c n lim Ä_ n lim

Ä_ n n ‹

œ n n b È

n # c

n

œ 1 b É

" n œ " # Ê converges 88. n lim Ä_ n lim Ä_ n lim

Ä_

"

1

c

" È n # c 1 c È n # b n œ Š È n # c 1 n n È n c 1 "

c È # b ‹Š È È n n

n # È È

# c 1 b n # b È #

b b

n ‹

œ n # c 1 b b

c 1 c n

n #

n

œ n lim Ä _

É 1 c n " #

b É

1

b

" n ˆ c " n

c

1

‰

œ c 2 Ê converges 89. n lim Ä_ " n '

1

n

" x dx œ n lim Ä_ ln n

n œ n lim Ä_

" n

œ 0 Ê


90. n lim Ä_ '

1

n x " p dx œ n lim Ä_ ’ 1 c " p x p " c 1 “ n 1

œ n lim Ä_

" 1 c p ˆ n p " c

1

c 1 ‰

œ p c "

1 if p 1 Ê converges

91. Since a n converges Ê n lim Ä_ a n œ L Ê n lim Ä_ a n b

1

œ n lim Ä_

1 b 72 a n

Ê L œ 72 1 b

L

Ê L a 1 b L b

œ 72 Ê L 2 b L c 72 œ

0 Ê L œ c 9 or L œ 8; since a n

0 for n 1 Ê L œ 8 92. Since a n converges Ê n lim Ä_ a n œ L Ê n lim Ä_ a n b

1

œ n lim Ä_

a a n n

b

6 b 2 Ê L œ L b

6 L b 2

Ê L a L b 2 b

œ L b 6 Ê L 2 b L c 6 œ

0 Ê L œ c 3 or L œ 2; since a n


1 œ n lim Ä_

È 8 b 2a n

Ê L œ È

8 b 2L Ê L 2 c 2L c 8 œ 0 Ê L œ c

2 or L œ 4; since a n


1 œ n lim Ä_

È 8 b 2a n

Ê L œ È

8 b 2L Ê L 2 c 2L c 8 œ 0 Ê L œ c

2 or L œ 4; since a n


1 œ n lim Ä_

È 5a n

Ê L œ È

5L Ê L 2 c 5L œ 0 Ê L œ 0 or L œ

5; since

a n

0 for n 1 Ê L œ

5

96. Since a n converges Ê n lim Ä_ a n œ L Ê n lim Ä_ a n b

1 œ n lim Ä_

ˆ 12 c ‰È a n

Ê L œ Š 12 c È

L ‹

Ê L 2 c 25L b 144 œ

0

Ê L œ 9 or L œ 16; since 12 c È

a n 12 for n 1 Ê L œ 9 97. a n b 1 œ 2 b a 1 n , n 1, a 1 œ 2. Since a n converges Ê n lim Ä_ a n œ L Ê n lim Ä_ a n b 1

œ n lim Ä_

Š 2 b a 1 n

‹

Ê L œ 2

b

1 L Ê L 2

c 2L c 1 œ 0 Ê L œ 1 „ È 2; since a n

0 for n 1 Ê L œ 1 b È 2 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

574 Chapter 10 Infinite Sequences and Series98. a n b 1 œ È 1 b a n , n 1, a 1 œ È 1. Since a n converges Ê n lim Ä_ a n œ L Ê n lim Ä_ a n b

1 œ n lim Ä_

È 1 b a n Ê L œ È

1 b L

Ê L 2 c L c 1 œ 0 Ê L œ 1 „ 2 È 5 ; since a n

0 for n 1 Ê L œ 1 b

2

È

5

99. 1, 1, 2, 4, 8, 16, 32, á œ 1, 2 ! , 2 " , 2 # , 2 $ , 2 % , 2 &

, á Ê x " œ 1 and x n

œ 2 n 2c for n 2 100. (a) 1 # c 2(1) # œ c 1, 3 # c 2(2) # œ 1; let f(a ß b) œ (a b 2b) # c 2(a b b) # œ a # b 4ab b 4b # c 2a # c 4ab c

2b # œ 2b # c a # ; a # c 2b # œc 1 Ê f(a ß b) œ 2b # c a # œ 1; a # c 2b # œ 1 Ê f(a ß b) œ 2b # c a # œc 1 (b) r # n c 2 œ ˆ a a b b 2 b b c c c b b

2b b ‰ # c œ a # 4ab 4b # (a „" r 2 "

2a # b) # 4ab 2b

#

œ c a a # c

2b # b (a b) # œ y #

n Ê n

œ Ê „ Š y n

‹ #

In the first and second fractions, y n

n. Let a b represent the (n c 1)th fraction where a b

1 and b n c 1 for n a positive integer 3. Now the nth fraction is a b

2b a b b

and a b b 2b 2n c 2 n Ê y n n. Thus, n lim Ä _ r n

œ

È

2.

101. (a) f(x) œ x # c 2; the sequence converges to 1.414213562 È

2 (b) f(x) œ tan (x) c 1; the sequence converges to 0.7853981635 1 4 (c) f(x) œ e x ; the sequence 1, 0, c 1, c 2, c 3, c 4, c 5, á diverges 102. (a) n

lim Ä _

nf ˆ ‰" n œ ? x lim Ä! b f( ? x) ? x œ ? x

lim Ä!

b

f(0 b ?

x) c f(0) ?

x œ f w

(0), where ?

x

œ

" n (b) n lim Ä _

n tan c" ˆ " n ‰

œ f w (0) œ " 1 b

0

#

œ 1, f(x) œ

tan c" x

(c) n lim Ä _

n a e 1 Î n c 1 b œ f w (0) œ e !

œ 1, f(x) œ e x c 1

(d) n lim Ä _

n ln ˆ 1 b 2 n ‰ œ f w

(0) œ 2

1 b

2(0)

œ 2, f(x) œ ln(1 b 2x)

103. (a) If a œ 2n b 1, then b œÚ a # # ÛœÚ 4n # b # 4n b 1 ÛœÚ 2n # b 2n b " # Ûœ 2n # b 2n, c œÜ a # #

ÝœÜ 2n #

b 2n b " # Ý œ 2n # b 2n b 1 and a # b b # œ (2n b 1) # b a 2n # b 2n b

#

œ 4n # b 4n b 1 b 4n % b 8n $ b 4n # œ 4n % b 8n $ b 8n # b 4n b 1 œ a 2n # b 2n b 1 b

#

œ c # . (b) a lim Ä_ Ú Ü a # a # # # Û Ý œ a lim Ä_ 2n #

b

2n 2n #

b 2n b

1

œ 1 or a lim Ä_ Ú Ü a

#

a

#

# #

Û Ý

œ a

lim Ä_

sin ) œ ) Ä lim 1

Î

2

sin )

œ 1

104. (a) n lim Ä_ (2n 1 ) 1 Î a 2n b

œ n lim Ä_ exp ˆ ln 2n 2n

1

‰ œ n lim Ä_ exp : Š 2n 2 # 1 1

‹

;

œ n lim Ä_

exp ˆ #

" n

‰

œ e !

œ 1;

n! ˆ n e ‰ È n 2n 1 , Stirlings approximation Ê È n n! ˆ n e ‰ (2n 1 ) 1 Î a 2n b

n e for large values of n (b) n È

n

n! 40 15.76852702 n

14.71517765

e

50 19.48325423 18.393

97206 60 23.19189561 22.07276647

105. (a) n lim Ä_ ln n

n c œ n lim Ä_ cn ˆ ‰"

n c c

1 œ n lim Ä_

" cn c

œ

0

(b) For all % 0, there exists an N œ e cÐ ln % ÑÎ c such that n e cÐ ln % ÑÎ c Ê ln n c ln c

%

Ê ln n c ln ˆ " %

‰

Ê n c

" % Ê " n c % Ê n " c c 0

% Ê n

Ä lim _

" n

c

œ

0

106. Let {a n } and {b n } be sequences both converging to L. Define {c n } by c 2n œ b n and c 2n c

1 œ

a n

, where n œ 1, 2, 3, á . For all % 0 there exists N " such that when n N " then k a n

c L k

%

and there exists N # such that when n N # then k b n c L k % . If n 1 b 2max{N " ß N #

}, then k c n c L k

%

, so {c n } converges to L. Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

Section 10.1 Sequences 575107. n lim Ä_ n 1 Î n

œ n lim Ä_ exp ˆ " n ln n ‰ œ n lim Ä_

exp ˆ " n

‰

œ e ! œ 1

108. n lim Ä_ x 1 Î n

œ n lim Ä_

exp ˆ " n

ln x ‰

œ e ! œ 1, because x remains fixed while n gets large

109. Assume the hypotheses of the theorem and let % be a positive number. For all %

there exists a N " such that when n N " then k a n c L k

% Ê c % a n c L % Ê L c %

a n

, and there exists a N # such that when n N # then k c n c L k

% Ê c % c n c L % Ê c n

L b %

. If n max{N " ß

N # }, then L c % a n Ÿ b n Ÿ c n L b % Ê k b n c L k

%

Ê n lim Ä _

b n œ L. 110. Let % ! . We have f continuous at L Ê there exists $ so that k x c L k $ Ê k f(x) c f(L) k

% . Also, a n Ä L Ê there exists N so that for n N k a n c L k $ . Thus for n N, k f(a n ) c f(L) k

%

Ê f(a n ) Ä f(L). 111. a n b

1 a n

Ê 3(n b 1) b

1 (n b 1) b 1 3n b 1 n b 1 Ê 3n b 4 n b# 3n b 1

n b

1

Ê 3n # b 3n b 4n b 4 3n # b 6n b n b

2 Ê 4 2; the steps are reversible so the sequence is nondecreasing; 3n

n b

b"

1 3 Ê 3n b 1 3n b 3 Ê 1 3; the steps are reversible so the sequence is bounded above by 3 112. a n b

1 a

n

Ê (2(n ((n b b 1) 1) b b 1)! 3)! (2n (n b b 1)! 3)! Ê (2n (n b b 2)! 5)! (2n (n b b 1)! 3)! Ê (2n (2n b b 5)! 3)!

(n (n b b

2)! 1)! Ê (2n b 5)(2n b 4) n b 2; the steps are reversible so the sequence is nondecreasing; the sequence is not bounded since (2n (n b

b

1)!

3)!

œ (2n b 3)(2n b 2) â (n b

2) can become as large as we please

113. a n b

1 Ÿ a n

Ê 2 n b 1 3 n b 1 (n b

1)! Ÿ 2 n 3 n n! Ê 2 n b 1 3

n b

1 2 n 3 n

Ÿ (n b n!

1)!

Ê 2 †

3 Ÿ n b

1 which is true for n 5; the steps are

reversible so the sequence is decreasing after a & , but it is not nondecreasing for all its terms; a " œ 6, a #

œ

18, a $ œ 36, a % œ 54, a &

œ 324

5

œ 64.8 Ê the sequence is bounded from above by 64.8

114. a n b

1 a n

Ê 2 c n b 2 1 c " # n b 1 2 c 2 n c " # n Ê 2 n c n b 2 1 " # n b 1 c " # n Ê 2

n(n b 1)

c

" # n b

1 ; the steps are reversible so the sequence is nondecreasing; 2 c 2 n

c #n " Ÿ 2 Ê the sequence is bounded from above 115. a n

œ 1 c " n converges because " n

Ä 0 by Example 1; also it is a nondecreasing sequence bounded above by 1

116. a n

œ n c " n diverges because n Ä _ and " n

Ä 0 by Example 1, so the sequence is unbounded

117. a n

œ 2 n

c 1 2 n œ 1 c " # n and 0 " # n " n ; since " n

Ä 0 (by Example 1) Ê " #

n

Ä

0, the sequence converges; also it is a nondecreasing sequence bounded above by 1

118. a n

œ 2 n

c 1 3 n œ ˆ 2

3 ‰

n

c " 3 n

; the sequence converges to !

by Theorem 5, #4

119. a n œ a ( c 1) n b 1 bˆ n b n 1

‰ diverges because a n œ 0 for n odd, while for n even a n

œ 2 ˆ 1 b " n

‰ converges to 2; it diverges by definition of divergence

120. x n œ max{cos 1 ß cos 2 ß cos 3 ßá ß cos n} and x n b

1 œ max{cos 1 ß cos 2 ß cos 3 ßá ß cos(n b 1)} x n with x n

Ÿ

1 so the sequence is nondecreasing and bounded above by 1 Ê the sequence converges. 121. a n a n b

1

Í 1 b È 2n

È n

"b È

2(n b

1) È

n b

1

Í È n b 1 b È 2n # b 2n ÈÈ n b 2n #

b 2n Í È n b

1 È

n

and 1 b

È

È n

2n

È 2 ; thus the sequence is nonincreasing and bounded below by È

2 Ê

it converges


576 Chapter 10 Infinite Sequences and Series122. a n a n b

1

Í n b 1 n (n b 1)

b" n b

1 Í n # b 2n b 1 n #

b 2n Í

1 0 and n b n

1

1; thus the sequence is nonincreasing and bounded below by 1 Ê it converges 123. 4 n b 1 b

3 n 4 n 4 b ˆ ‰3

4

œ n

4 b ˆ 3 4 ‰ n so a n a n b

1

Í 4 b ˆ 3 4 ‰ n 4 b ˆ 3 4 ‰ n b" Í ˆ 3 4 ‰ n ˆ 3 4 ‰

n b

1 Í 1 4; thus the sequence is nonincreasing and bounded below by 4 Ê it converges 3 4 and 124. a " œ 1, a # œ 2 c 3, a $ œ 2(2 c 3) c 3 œ 2 # c a 2 2 c" b † 3, a %

œ 2 a 2 # c a 2 2 c" b †

3 b c 3 œ 2 $ c a 2 $ c 1 b 3, a &

œ 2 c 2 $ c a 2 $ c 1 b

3 d c 3 œ 2 % c a 2 % c 1 b 3, á , a n

œ 2 n c" c a 2 n c" c 1 b 3 œ 2 n c" c 3 †

2 n c 1 b 3 œ 2 n c 1 (1 c 3) b 3 œc 2 n b 3; a n a n b

1

Í c 2 n b 3 c 2 n b 1 b 3 Í c 2 n c 2 n b

1 Í 1 Ÿ 2 so the sequence is nonincreasing but not bounded below and therefore diverges

125. Let 0 M 1 and let N be an integer greater than M 1 c M . Then n N Ê n M 1 c M Ê n c nM M Ê n M b nM Ê n M(n b 1) Ê n

n 1b M. 126. Since M " is a least upper bound and M # is an upper bound, M " Ÿ

M # . Since M # is a least upper bound and M

" is an upper bound, M # Ÿ M " . We conclude that M " œ

M #

so the least upper bound is unique.

127. The sequence a n

œ 1 b ( c"

) n # is the sequence " # , 3 # , "

# , 3 # , á

. This sequence is bounded above by 3 #

, but it clearly does not converge, by definition of convergence.

128. Let L be the limit of the convergent sequence {a n

}. Then by definition of convergence, for % #

there corresponds an N such that for all m and n, m N Ê k a m c L k % # and n N Ê k a n

c L k

# % . Now k a m c a n k œ k a m c L b L c a n k Ÿ k a m c L k b k L c a n k

% # b % #

œ % whenever m N and n N. 129. Given an % 0, by definition of convergence there corresponds an N such that for all n N, k L " c a n k % and k L # c a n k % . Now k L # c L " k œ k L # c a n b a n c L " k Ÿ k L # c a n k b k a n

c L "

k

% b % œ

2 % . k L # c L "

k

2 % says that the difference between two fixed values is smaller than any positive number 2 % . The only nonnegative number smaller than every positive number is 0, so k L " c L # k

œ 0 or L " œ

L # . 130. Let k(n) and i(n) be two order-preserving functions whose domains are the set of positive integers and whose ranges are a subset of the positive integers. Consider the two subsequences a kn Ð Ñ and a in Ð Ñ , where a kn Ð Ñ Ä

L "

, a in Ð Ñ Ä L # and L " Á L # . Thus a kn Ð Ñ c a in Ð Ñ Ä k L " c L #

k

0. So there does not exist N such that for all m, n N Ê k a m c a n k

%

. So by Exercise 128, the sequence Ö a n × is not convergent and hence diverges. 131. a 2k Ä L Í given an % 0

there corresponds an N " such that c 2k N "

Ê k a 2k

c L k

%

d . Similarly, a 2k b 1 Ä L Í c 2k b 1 N # Ê k a 2k b 1 c L k % d . Let N œ max{N " ß N #

}. Then n N Ê k a n c L k %

whether n is even or odd, and hence a n

Ä

L.

132. Assume a n Ä 0. This implies that given an % 0 there corresponds an N such that n N Ê k a n

c 0

k % Ê k a n k % Ê k k a n k k % Ê k k a n k

c 0 k %

Ê k a n k Ä 0. On the other hand, assume k a n k Ä 0. This implies that given an % 0 there corresponds an N such that for n N, k k a n k c 0 k % Ê k k a n k

k % Ê k a n k % Ê k a n c 0 k

%

Ê a n Ä 0. 133. (a) f(x) œ x # c a Ê f w

(x) œ 2x Ê x n b 1 œ x n c x # n # x c n a Ê x n b

1

œ 2x # n c 2x a x n # n c a b

œ x #

n 2x

b

n

a

œ ˆ x n

b #

x

a n

‰

(b) x " œ 2, x # œ 1.75, x $ œ 1.732142857, x % œ 1.73205081, x &

œ

1.732050808; we are finding the positive number where x # c 3 œ 0; that is, where x # œ 3, x 0, or where x œ

Section 10.2 Infinite Series 577134. x " œ 1, x # œ 1 b cos (1) œ 1.540302306, x $

œ 1.540302306 b cos (1 b cos (1)) œ

1.570791601, x %

œ 1.570791601 b cos (1.570791601) œ 1.570796327 œ 1 #

to 9 decimal places. After a few steps, the arc a x n c 1 b and line segment cos a x n c

1 b are nearly the same as the quarter circle. 135-146. Example CAS Commands:

Mathematica

: (sequence functions may vary): Clear[a, n] a[n_]; = n1 / n first25= Table[N[a[n]],{n, 1, 25}] Limit[a[n], n Ä 8] Mathematica: (sequence functions may vary):

Clear[a, n] a[n_]; = n1 / n first25= Table[N[a[n]],{n, 1, 25}] Limit[a[n], n Ä 8] The last command (Limit) will not always work in Mathematica. You could also explore the limit by enlarging your table to more than the first 25 values. If you know the limit (1 in the above example), to determine how far to go to have all further terms within 0.01 of the limit, do the following.

Clear[minN, lim] lim= 1 Do[{diff=Abs[a[n] c lim], If[diff < .01, {minN= n, Abort[]}]}, {n, 2, 1000}] minN For sequences that are given recursively, the following code is suggested. The portion of the command a[n_]:=a[n] stores the elements of the sequence and helps to streamline computation.

Clear[a, n] a[1]= 1; a[n_]; = a[n]= a[n c 1] b (1/5) (n c 1) first25= Table[N[a[n]], {n, 1, 25}] The limit command does not work in this case, but the limit can be observed as 1.25.

Clear[minN, lim] lim= 1.25 Do[{diff=Abs[a[n] c lim], If[diff < .01, {minN= n, Abort[]}]}, {n, 2, 1000}] minN

10.2 INFINITE SERIES

1. s n œ a a 1 c

r

n

b (1 c

r)

œ 2 ˆ 1 1

c c

ˆ ‰ˆ " 3 " 3

‰ ‰

n

Ê n lim Ä _

s n

œ 2 1

c

ˆ " 3

‰

œ

3

2. s n a a 1 c

r

n b (1 c r) œ œ ˆ 100 9 ‰ˆ 1 c 1

c ˆ 100 " ˆ 100 ‰ "

‰

n

‰ Ê n lim Ä _

s

n

œ 1

c ˆ 100 ˆ 9

100 ‰ " ‰

œ

" 11

3. s n œ a a 1 c

r

n

b (1 c

r) œ 1 1

c c ˆ ˆ c c

" #

" # ‰ ‰ n

Ê n lim Ä _

s

n

œ " ˆ 3 #

‰

œ

2 3

4. s n

œ 1 c ( c

2)

n 1 c ( c

2)

, a geometric series where k r k

1 Ê

divergence

5. " (n b 1)(n b# ) œ n b " 1 c n b# " Ê s n œ ˆ " # c 3 " ‰ b ˆ " 3 c " 4 ‰ bá b ˆ n b " 1 c n b# " ‰

œ " # c n

b# " Ê n lim Ä _

s n

œ " # Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

578 Chapter 10 Infinite Sequences and Series6. 5 n(n b 1) œ 5 n c n b 5 1 Ê s n

œ ˆ 5 c 5 2 ‰ b ˆ 5 2 c 5 3 ‰ b ˆ 5 3 c 5 4 ‰ bá b ˆ n c 5 1 c 5 n ‰ b ˆ 5 n c n b 5 1 ‰

œ 5 c n b 5

1 Ê n lim Ä _

s n

œ 5 7. 1 c " 4 b 16 " c 64 " b á , the sum of this geometric series is 1 c " ˆ c " 4 ‰ œ "

1

b ˆ " 4

‰ œ 4 5 8. " 16 b 64 " b 256 " b á , the sum of this geometric series is 1

ˆ c 16 " ˆ ‰ " 4

‰

œ

" 1 # 9. 7 4 b 16 7 b 64 7 b á , the sum of this geometric series is 1

c ˆ 7 4 ˆ ‰ " 4

‰

œ

7 3 10. 5 c 5 4 b 16 5 c 64 5 b á , the sum of this geometric series is 5 1 c cˆ " 4 ‰ œ 4 11. (5 b 1) b ˆ 5 # b " 3 ‰ b ˆ 5 4 b " 9 ‰ b ˆ 5

8 b # " 7 ‰ b á , is the sum of two geometric series; the sum is 5 1 c ˆ " #

‰ b "

1 c

ˆ " 3

‰ œ 10 b 3 # œ

23 # 12. (5 c 1) b ˆ 5 # c " 3 ‰ b ˆ 5 4 c " 9 ‰ b ˆ 5

8 c # " 7 ‰ b á , is the difference of two geometric series; the sum is 5 1 c ˆ " #

‰ c "

1 c

ˆ " 3

‰ œ 10 c 3 # œ

17 # 13. (1 b 1) b ˆ 1 # c " 5 ‰ b ˆ 1 4 b 25 " ‰ b ˆ 1

8 c 1 " # 5 ‰ b á , is the sum of two geometric series; the sum is 1 1 c ˆ " #

‰ b "

1 b

ˆ " 5

‰ œ 2 b 5 6 œ

17 6 14. 2 b 4 5 b 25 8 b 125 16 bá œ 2 ˆ 1 b 2 5 b 25 4 b 125 8 bá ‰ ; the sum of this geometric series is 2 Š 1

c "

ˆ 2 5

‰ ‹ œ 10 3 15. Series is geometric with r œ 2 5 Ê 1 2 5 1 1 Ê Converges to 1 1

c 2 5

œ 5 3 16. Series is geometric with r œc 3 Ê 1 c 3 1 1 Ê Diverges 17. Series is geometric with r œ 1 8 Ê 1 1 8 1

1 Ê Converges to 1 8 1

c

1 8

œ 1 7 18. Series is geometric with r œc 2 3 Ê 1 c 2 3 1

1 Ê Converges to 1

c c

ˆ c 2 3

2 3

‰

œc 2 5 19. 0.23 œ ! _ n œ 0 23 100 ˆ 10 " # ‰ n œ Š 100 23 ‹ 1

c

ˆ 100

"

‰

œ 23 99 20. 0.234 œ ! _ n œ 0

234 1000 ˆ 10 "

$

‰

n

œ Š 1000 234

‹

1

c

Š 1000

"

‹

œ 234 999

21. 0.7 œ ! _ n œ 0 7 10 ˆ 10 " ‰ n œ Š 10 7 ‹ 1 c Š 10 " ‹ œ 7 9 22. 0.d œ ! _ n œ 0

d 10 ˆ 10 "

‰

n

œ Š 10 d

‹

1

c

Š 10 "

‹

œ d 9

23. 0.06 œ ! _ œn 0

ˆ 10 1 ‰ˆ 10 6 ‰ˆ 10 " ‰

n

œ Š 100

6

‹

1

c

Š 10 "

‹

œ 6

90 œ " 15

24. 1.414 œ 1 b ! _ œn 0

414 1000 ˆ 10 " $

‰

n

œ 1 b Š 1000 414

‹

1

c

Š 1000

"

‹

œ 1 b 414 999 œ "

413 999


ISMT12_C10_A

Documents

Transcript of ISMT12_C10_A