Thomas Calculus 13th [Solutions]

754 Chapter 10 Infinite Sequences and Series
11.
un
n 1
x n
n
n
1
lim 1 lim ! 1 x lim 1 1 for all x
n
u
n
( n 1)! x
n
n 1
(a) the radius is ; the series converges for all x
(b) the series converges absolutely for all x
(c) there are no values for which the series converges conditionally
12.
u n 1 n 1
n
x n
n n
n
lim 1 lim 1 3 x lim 1 for all x
1 3 ! 1
n
u
n
( n 1)! 3 x
n
n 1
(a) the radius is ; the series converges for all x
(b) the series converges absolutely for all x
(c) there are no values for which the series converges conditionally
13.
u 1 2 2
1
n n
n
x n n
u n 2n
n
n n
4 2 4 2 2 1 1 1
n n
1 4 x
n
1 4 2 2
lim 1 lim 1 x lim 4 x 1 x x ; when x 1
we have
n
4 1
2n
1
n 2 n
n 1 n 1
,
a divergent p-series when x 1
we have
2
p-series
(a) the radius is 1 2 ; the interval of convergence is 1 x 1
2 2
(b) the interval of absolute convergence is
1 x 1
2 2
(c) there are no values for which the series converges conditionally
n
4 1
2n
1
n 2 n
n 1 n 1
,
a divergent
2
14.
lim 1
u 2 2
1
1 lim ( 1) n
n
n
x n 3 1 1 lim n 1
2 1 2 3
1 1 2 4;
n
n
n
n
u
n ( n 1) 3 ( x 1) n 3( n 1)
x x x when x 2 we
n
( 3) ( 1)
n 3 n
n 1 n 1
n
(3) 1
n 3 n
n 1 n 1
have , an absolutely convergent series; when x 4 we have ,
2 n
2
2 n
2
absolutely convergent series.
(a) the radius is 3; the interval of convergence is 2 x 4
(b) the interval of absolute convergence is 2 x 4
(c) there are no values for which the series converges conditionally
n
an
15.
u
n
u
n ( n 1) 3 x n n 2n
4
lim
1 2
2
1
1 lim n
n
x n 3 1 lim n 3
1 1 1 1;
2 n
2
n
x x x when x 1
we have
n 1
n
( 1)
2
n 3
,
a conditionally convergent series; when x 1 we have
(a) the radius is 1; the interval of convergence is 1 x 1
(b) the interval of absolute convergence is 1 x 1
(c) the series converges conditionally at x 1
n
1
1
2
n
3
,
a divergent series
16.
u
n
u
n ( n 1) 3 x n n 2n
4
lim
1 2
2
1
1 lim n
n
x n 3 1 lim n 3
1 1 1 1;
2 n
2
n
x x x when x 1
we have
n 1
1
2
n
3
,
a divergent series; when x 1 we have
(a) the radius is 1; the interval of convergence is 1 x 1
(b) the interval of absolute convergence is 1 x 1
(c) the series converges conditionally at x 1
n
1
( 1)
n
2
n
3
,
a conditionally convergent series
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