Thomas Calculus 13th [Solutions]

718 Chapter 10 Infinite Sequences and Series
74. a 1, r 1 ; converges to
2
x
1
1
1
x
2
2
x
2
x
1
for 1 1 or x 1
2
x
75. a 1, r ( x 1); converges to 1 1
1 ( x 1) 2 x for x 1 1 or 2 x 0
76. a 1, r 3 x ; converges to 1 2
2
1
3 x x 1
2
for 3 x 1 or 1 x 5
2
77. a 1, r sin x ; converges to 1
1 sin x for x (2k 1) , k an integer
2
78. a 1, r ln x ; converges to 1
1 ln x
for ln x 1 or 1
e x e
79. (a) 1
( 4)( 5)
n 2 n n
(b) 1
( 2)( 3)
n 0 n n
(c) 1
( 3)( 2)
n 5 n n
80. (a) 5
( 2)( 3)
n 1 n n
(b) 5
( 2)( 1)
n 3 n n
(c) 5
( 19)( 18)
n 20 n n
81. (a) one example is
(b) one example is
1 1 1 1
2 4 8 16 1
1
2
1
2
3 3 3 3
2 4 8 16 1
2
(c) one example is 1 1 1 1 1 1 0
2 4 8 16 1
3
2
1
2
1
1
1
2
3
82. The series
number.
n
1
n 1
k is a geometric series whose sum is
2
0
1
k
2
1
2
k where k can be any positive or negative
83. Let 1
n
an
b n . Then 1
n
a
2
n b n
1, while
2
n 1 n 1 n 1
n
an
(1)
bn
1 n 1
diverges.
84. Let 1
n
an
b n . Then 1
n
a
2
n b n
1, while
2
( ) 1
n
a 1
nbn
AB
4 3
.
n 1 n 1 n 1
n 1 n 1
85. Let 1
n
a n and 1
n
b .
4 n Then 1
an
A a , 1
2
n B b
3
n and 1
bn
2
n 1 n 1
n 1 n 1
n
1 A .
B
86. Yes: 1 diverges. The reasoning:
a n
nth-Term Test.
a n converges a n 0 1 1
a a diverges by the
n
n
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