Page 314 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 11] INFINITE SERIES 305

11.139. Prove that the Ce ´ saro method of summability is regular. [Hint: See Page 278.]
3
2
2
11.140. Prove that the series 1 þ 2x þ 3x þ 4x þ þ nx n 1 þ converges to 1=ð1 xÞ for jxj < 1.
1 1
X X
n
11.141. A series a n is called Abel summable to S if S ¼ lim a n x exists. Prove that
n¼0 x!1 n¼0
1
X n
ð 1Þ ðn þ 1Þ is Abel summable to 1/4 and
ðaÞ
n¼0
n
1
X
is Abel summable to 1/8.
ð 1Þ ðn þ 1Þðn þ 2Þ
2
ðbÞ
n¼0
1
1 1
X X
11.142. Prove that the double series , where p is a constant, converges or diverges according as
2 2 p
m¼1 n¼1 ðm þ n Þ
p > 1or p @ 1, respectively.
e
e
ð 1 x u 1 1 2! 3! n 1 ðn 1Þ! ð 1 x u
n
11.143. (a)Prove that du ¼ 2 þ 3 4 þ ð 1Þ n þð 1Þ n! nþ1 du.
x u x x x x x x u
ð 1 x u 1 1 2! 3!
e
ðbÞ Use ðaÞ to prove that du 2 þ 3 4 þ
x u x x x x