Thomas Calculus 13th [Solutions]

756 Chapter 10 Infinite Sequences and Series
2 x 0; when x 2 we have
n
1
n
n 1
2 ( 1) ( 1) ,
a divergent series; when x 0 we have
n 1
n
2 ( 1) ,
a divergent series
(a) the radius is 1; the interval of convergence is 2 x 0
(b) the interval of absolute convergence is 2 x 0
(c) there are no values for which the series converges conditionally
22.
lim 1 2 2 1
un
1
1 lim ( 1) n 3 n ( x 2) n
3n 9 17 19
3( 1) 1 x 2 lim n
2
1 9 x 2 1 x
u 9 9
;
n n n
n
n n
n n ( 1) 3 ( x 2)
n
when x 17
we have
9
n 2n n
( 1) 3 1 1
,
3n
9 3n
n 1 n 1
2
( 1) n 3 n n
1 ( 1)
n
,
3n
9 3n
n 1 n 1
a conditionally convergent series.
(a) the radius is 1 ; the interval of convergence is
17 x 19
9 9 9
(b) the interval of absolute convergence is
(c) the series converges conditionally at x 19
9
17 x 19
9 9
a divergent series; when x 19
we have
9
23.
1
n 1 1
1
t
n
u
1 x
lim 1
n 1
n 1
t
t
lim 1 lim 1 1
e
1 1 1 1;
u
1
1
n n
lim 1
1
n
n
e
n
n
n
n n x
n 1
n
n
n
x x x x when x 1
we have ( 1) 1
1
, a divergent series by the nth-Term Test since lim 1
1 e 0; when x 1
we have 1
1
, a divergent series
n 1
n
n
(a) the radius is 1; the interval of convergence is 1 x 1
(b) the interval of absolute convergence is 1 x 1
(c) there are no values for which the series converges conditionally
n
n
n
24.
1
1
lim 1
1 lim n
un
ln ( n 1) x 1 lim n 1
x 1 x lim n
ln
1
1
1 x 1 1 x
u
1;
n
n
x n
n
n
n n n n
when x 1 we have
n
1
n
( 1) ln n , a divergent series by the nth-Term Test since lim ln n 0; when x 1
n
we have ln n , a divergent series
n 1
(a) the radius is 1; the interval of convergence is 1 x 1
(b) the interval of absolute convergence is 1 x 1
(c) there are no values for which the series converges conditionally
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