Thomas Calculus 13th [Solutions]

794 Chapter 10 Infinite Sequences and Series
80. The approximation sin x 6x
is better than
2
6 x
sin x x.
81.
n 1
2 5 8 (3n 1)(3n 2) x 2 4 6 (2 n) lim 1 x lim 3n
2 1 x 2 the radius of convergence is 2 2 4 6 (2 n)(2n 2) n
n
2 5 8 (3n
1) x
n
2n
2 3
3
82.
n 1
3 5 7 (2n 1)(2n 3)( x 1) 4 914 (5n 1)
lim 1 x lim 2n
3 1 x 5 the radius of convergence
4 9 14 (5 1)(5 4) n
n
n n 3 5 7 (2n
1) x
n
5n
4 2
is 5 2
n n n
83. ln 1 1 ln 1 1 ln 1 1 ln( k 1) ln k ln( k 1) ln k
2
k
k
k
k 2 k 2 k 2
ln 3 ln 2 ln1 ln 2 ln 4 ln 3 ln 2 ln 3 ln 5 ln 4 ln 3 ln 4 ln 6 ln 5 ln 4 ln 5
ln( n 1) ln n ln( n 1) ln n ln1 ln 2 ln( n 1) ln n after cancellation
n
ln 1 1 ln n 1 ln 1 1 lim ln n 1 ln 1 is the sum
2 2 2
k n
k 2n
2
k 2 k 2
n
n
n
84. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2
k 1 2 k 1 k 1 2 1 3 2 4 3 5 4 6 n 2 n n 1 n 1
k 2 k 2
2
1 1 1 1 1 1 3 1 1 1 3 n( n 1) 2( n 1) 2n 3n n 2 1 lim 1 3 1 1 3
2 1 2 n n 1 2 2 n n 1 2 2 n( n 1) 4 n( n 1) 2 2 2 n n 1 4
k
k 2
1
n
85. (a)
(b)
3 n 3
lim 1 4 7 (3 2)(3 1) (3 )! 3 (3 1)
3
(3 3)! x lim x
3n
n
n 1 4 7 (3n
2) x
n
(3n 1)(3n 2)(3n
3)
0 1
the radius of convergence is
1 4 7 (3n 2) 3n dy 1 4 7 (3n 2) 3n 1 d y 1 4 7 (3n
2) 3n
2
y 1
x x x
(3 n)! dx (3n 1)! 2
dx (3n
2)!
n 1 n 1 n 1
2
1 4 7 (3n
5) 3n
2 1 4 7 (3 2) 3
(3 3)! 1 n n
x x x x xy
n
(3 n)!
0 a 1 and b 0
n 2 n 1
86. (a)
(b)
x 2 x 2
2 2 2 2 2 3 2 3 4 5
x x ( x) x ( x) x ( x) x x x x ( 1) n x
n
1 x 1 ( x)
n 2
which converges absolutely for x 1
n n n
x 1 ( 1) x ( 1) which diverges
n 2 n 2
Copyright
2014 Pearson Education, Inc.