Page 311 - Schaum's Outline of Theory and Problems of Advanced Calculus

302 INFINITE SERIES [CHAP. 11

POWER SERIES
x 2 x 3 x 4
þ .
2 3 4
11.100. (a)Prove that lnð1 þ xÞ¼ x þ
1
1
1
ðbÞ Prove that ln 2 ¼ 1 þ þ :
2 3 4
1

2
3
Hint: Use the fact that ¼ 1 x þ x x þ and integrate.
1 þ x
11.101. Prove that sin 1 x ¼ x þ 1 x 3 þ 1 3 x 5 þ 1 3 5 x 7 þ , 1 @ x @ 1.
2 3 2 4 5 2 4 6 7
ð 1=2 ð 1 1 cos x
11.102. Evaluate (a) e x 2 dx; ðdÞ dx to 3 decimal places, justifying all steps.
0 0 x
Ans. ðaÞ 0:461; ðbÞ 0:486
11.103. Evaluate (a) sin 408; ðbÞ cos 658; ðcÞ tan 128 correct to 3 decimal places.
Ans: ðaÞ 0:643; ðbÞ 0:423; ðcÞ 0:213
11.104. Verify the expansions 4, 5, and 6 on Page 275.
11.105. By multiplying the series for sin x and cos x,verify that 2 sin x cos x ¼ sin 2x.
x 2 4x 4 31x 6 !
11.106. Show that e cos x ¼ e 1 þ þ ; 1 < x < 1.
2! 4! 6!
11.107. Obtain the expansions
x 3 x 5 x 7
ðaÞ tanh 1 x ¼ x þ þ þ þ 1 < x < 1
3 5 7
p ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x 3 1 3 x 5 1 3 5 x 7
2 1 @ x @ 1
ðbÞ lnðx þ x þ 1 Þ¼ x þ þ
2 3 2 4 5 2 4 6 7
1=x 2
e x 6¼ 0
11.108. Let f ðxÞ¼ .Prove that the formal Taylor series about x ¼ 0corresponding to f ðxÞ exists
0 x ¼ 0
but that it does not converge to the given function for any x 6¼ 0.
11.109. Prove that
1 1 1

lnð1 þ xÞ 2 3
1 þ x 2 2 3
ðaÞ ¼ x 1 þ x þ 1 þ þ x for 1 < x < 1
3 4
1 2x 1 1 2x
2 2
ðbÞflnð1 þ xÞg ¼ x 1 þ þ 1 þ þ for 1 < x @ 1
2 3 2 3 4
MISCELLANEOUS PROBLEMS
11.110. Prove that the series for J p ðxÞ converges (a)for all x,(b) absolutely and uniformly in any finite interval.
d d 2p
p
p
11.111. Prove that (a) fJ 0 ðxÞg ¼ J 1 ðxÞ; ðbÞ fx J p ðxÞg ¼ x J p 1 ðxÞ; ðcÞ J pþ1 ðxÞ¼ J p ðxÞ J p 1 ðxÞ.
dx dx x
11.112. Assuming that the result of Problem 11.111(c) holds for p ¼ 0; 1; 2; .. . ; prove that
n
(a) J 1 ðxÞ¼ J 1 ðxÞ; ðbÞ J 2 ðxÞ¼ J 2 ðxÞ; ðcÞ J n ðxÞ¼ ð 1Þ J n ðxÞ; n ¼ 1; 2; 3; .. . :
1
X
p
11.113. Prove that e 1=2xðt 1=tÞ ¼ J p ðxÞ t .
p¼ 1 xt=2 x=2t
[Hint: Write the left side as e e ,expand and use Problem 11.112.]