Page 295 - Schaum's Outline of Theory and Problems of Advanced Calculus

286 INFINITE SERIES [CHAP. 11

SERIES OF FUNCTIONS
11.25. For what values of x do the following series converge?
1 n 1 1 n 1 2n 1 1 1 n
X x X ð 1Þ x X n X nðx 1Þ
; ; n!ðx aÞ ; :
ðaÞ n ðbÞ ðcÞ ðdÞ n
n 3 ð2n 1Þ!
n¼1 n¼1 n¼1 n¼1 2 ð3n 1Þ
x n 1
. Assuming x 6¼ 0 (if x ¼ 0the series converges), we have
(a) u n ¼ n
n 3
n n
x n 3 n jxj
lim u nþ1 ¼ lim ¼ lim
n!1 ðn þ 1Þ 3 nþ1 n 1 jxj¼ 3

n!1 u n x n!1 3ðn þ 1Þ
Then the series converges if jxj < 1, and diverges if jxj > 1. If jxj ¼ 1, i.e., x ¼ 3, the test fails.
3 3 3
1 1 1
1 1
X X
If x ¼ 3the series becomes ¼ , which diverges.
3n 3 n
n¼1 n¼1
n 1 n 1
1 1
X ð 1Þ 1 X ð 1Þ
If x ¼ 3the series becomes ¼ , which converges.
3n 3 n
n¼1 n¼1
Then the interval of convergence is 3 @ x < 3. The series diverges outisde this interval.
Note that the series converges absolutely for 3 < x < 3. At x ¼ 3the series converges con-
ditionally.
n 1 2n 1
. Then
ð 1Þ x
(b)Proceed as in part (a)with u n ¼
ð2n 1Þ!

n 2nþ1
ð 1Þ x ð2n 1Þ!
2
lim u nþ1 ð2n 1Þ! ¼ lim x
n!1 ð2n þ 1Þ!
x
¼ lim n!1 ð2n þ 1Þ!
n!1 u n n 1 2n 1
ð 1Þ
ð2n 1Þ! x 2
2
¼ lim x ¼ lim ¼ 0
n!1 ð2n þ 1Þð2nÞð2n 1Þ! n!1 ð2n þ 1Þð2nÞ
Then the series converges (absolutely) for all x, i.e., the interval of (absolute) convergence is
1 < x < 1.

nþ1

n

u n ¼ n!ðx aÞ ; lim u nþ1 ðn þ 1Þ!ðx aÞ ¼ lim ðn þ 1Þjx aj:

ðcÞ ¼ lim n

n!1 u n n!1 n!1
n!ðx aÞ
This limit is infinite if x 6¼ a. Then the series converges only for x ¼ a.
n nþ1
: Then
nðx 1Þ ðn þ 1Þðx 1Þ
n 2 nþ1
ðdÞ u n ¼ ; u nþ1 ¼
2 ð3n 1Þ ð3n þ 2Þ

ðn þ 1Þð3n 1Þðx 1Þ x 1 jx 1j
lim u nþ1 ¼ lim ¼ ¼

n!1 u n n!1 2 2
2nð3n þ 2Þ
Thus, the series converges for jx 1j < 2 and diverges for jx 1j > 2.
The test fails for jx 1j¼ 2, i.e., x 1 ¼ 2or x ¼ 3 and x ¼ 1.
1
X n
For x ¼ 3the series becomes , which diverges since the nth term does not approach zero.
3n 1
n¼1
1 n
X ð 1Þ n
For x ¼ 1the series becomes , which also diverges since the nth term does not
approach zero. n¼1 3n 1
Then the series converges only for jx 1j < 2, i.e., 2 < x 1 < 2or 1 < x < 3.