Page 304 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 11] INFINITE SERIES 295

Adding,
n pþ2nþ2
1 ð 1Þ x
2 2 X
2 00 0
2 n!ðn þ pÞ!
x y þ xy þðx p Þy ¼ pþ2n
n¼0
ð 1Þ ½ p þð p þ 2nÞþð p þ 2nÞð p þ 2n 1ފx
1 n 2 pþ2n
X
2 n!ðn þ pÞ!
þ pþ2n
n¼0
1 n pþ2nþ2 1 n pþ2n
X ð 1Þ x X ð 1Þ ½4nðn þ pފx
2 n!ðn þ pÞ! 2 n!ðn þ pÞ!
¼ pþ2n þ pþ2n
n¼0 n¼0
n 1 pþ2n n pþ2n
1 1
X x X ð 1Þ 4x
ð 1Þ
2 ðn 1Þ!ðn 1 þ pÞ! 2 ðn 1Þ!ðn þ p 1Þ!
¼ pþ2n 2 þ pþ2n
n¼1 n¼1
1 n pþ2n 1 n pþ2n
X ð 1Þ 4x X ð 1Þ 4x
2 ðn 1Þ!ðn þ p 1Þ! 2 ðn 1Þ!ðn þ p 1Þ!
¼ pþ2n þ pþ2n
n¼1 n¼1
¼ 0
n 1
1
X z
11.47. Test for convergence the complex power series 3 n 1 .
n 3
n¼1

n 3 3
z n 3 n 1 n
jzj jzj
Since lim u nþ1 ¼ lim ,the series converges for < 1,

¼ lim 3 n n 1 3 jzj¼
n!1 u n n!1 ðn þ 1Þ 3 z n!1 3ðn þ 1Þ 3 3

i.e., jzj < 3, and diverges for jzj > 3.
n 1
1 1
X jzj X 1
For jzj¼ 3, the series of absolute values is ¼ ,sothat the series is absolutely
3
n 3 n 1 n 3
n¼1 n¼1
convergent and thus convergent for jzj¼ 3.
Thus, the series converges within and on the circle jzj¼ 3.
x
11.48. Assuming the power series for e holds for complex numbers, show that
ix
e ¼ cos x þ i sin x
z 2 z 3
z
2! 3!
Letting z ¼ ix in e ¼ 1 þ z þ þ þ ; we have
! !
3 3
2 2
i x i x x 2 x 4 x 3 x 5
ix
2! 3! 2! 4! 3! 5!
e ¼ 1 þ ix þ þ þ ¼ 1 þ þ ix þ
¼ cos x þ i sin x
Similarly, e ix ¼ cos x i sin x. The results are called Euler’s identities.
1 1 1 1

11.49. Prove that lim 1 þ þ þ þ þ ln n exists.
2 3 4 n
n!1
Letting f ðxÞ¼ 1=x in (1), Problem 11.11, we find
1 1 1 1 1 1 1 1
4
2 þ þ þ þ M @ ln M @ 1 þ þ þ þ þ M 1
2
3
4
3
from which we have on replacing M by n,
1 1 1 1 1
@ 1 þ þ þ þ þ ln n @ 1
n 2 3 4 n
1 1 1 1
Thus, the sequence S n ¼ 1 þ þ þ þ þ ln n is bounded by 0 and 1.
2 3 4 n