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c all x 6 0;
d x > 0;
e x @ 0
Prove that
1 X 1 3 5 2n 1 n 1
2 4 6 2n
xn converges for 1 @ x < 1.
UNIFORM CONVERGENCE 11.92. By use of the de nition, investigate the uniform convergence of the series
1 X n 1
! 1 : 1 nx Ans. Not uniformly convergent in any interval which includes x 0; uniformly convergent in any other interval. Hint: Resolve the nth term into partial fractions and show that the nth partial sum is Sn x 1 11.93. 11.94. Work Problem 11.30 directly by rst obtaining Sn x . Investigate by any method the convergence and uniform convergence of the series: a
1 Xxn n 1
x 1 n 1 x 1 nx
1 X sin2 nx n 1
2 1
x ; x A 0: 1 x n n 1
Ans. (a) conv. for jxj < 3; unif. conv. for jxj @ r < 3. (b) unif. conv. for all x. (c) conv. for x A 0; not unif. conv. for x A 0, but unif. conv. for x A r > 0. If F x
1 X sin nx , prove that: n3 n 1 (a) F x is continuous for all x, (b) lim F x 0; x!0
c F 0 x
1 X cos nx n 1
is continous everywhere.
 11.96. Prove that
 cos 2x cos 4x cos 6x dx 0. 1 3 3 5 5 7
1 X sin nx has derivatives of all orders for any real x. sinh n n 1
Prove that F x
Examine the sequence un x 1
1 ; n 1; 2; 3; . . . ; for uniform convergence. 1 x2n
Prove that lim
n!1 0
dx 1 e 1 . 1 x=n n
INFINITE SERIES
[CHAP. 11
POWER SERIES
11.100. (a) Prove that ln 1 x x
x2 x3 x4 . 2 3 4 1 1 1 b Prove that ln 2 1 2 3 4 : ! 1 Hint: Use the fact that 1 x x2 x3 and integrate. 1 x
11.101. Prove that sin 1 x x
1 x3 1 3 x5 1 3 5 x7 , 1 @ x @ 1. 2 3 2 4 5 2 4 6 7 1 1=2 2 1 cos x dx to 3 decimal places, justifying all steps. e x dx; d 11.102. Evaluate (a) x 0 0 Ans. a 0:461; b 0:486
11.103. Evaluate (a) sin 408; b cos 658; c tan 128 correct to 3 decimal places. Ans: a 0:643; b 0:423; c 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 sin 2x.
cos x
11.106. Show that e
! x2 4x4 31x6 ; 1 < x < 1. e 1 2! 4! 6!
11.107. Obtain the expansions a tanh 1 x b ln x x x3 x5 x7 3 5 7 1<x<1
p 1 x3 1 3 x5 1 3 5 x7 1 @ x @ 1 x2 1 x 2 3 2 4 5 2 4 6 7 & 1=x2 x 6 0 . Prove that the formal Taylor series about x 0 corresponding to f x exists 11.108. Let f x e 0 x 0 but that it does not converge to the given function for any x 6 0. 11.109. Prove that a ln 1 x 1 x
    1 1 1 x 1 x2 1 x3 2 2 3
for 1 < x < 1
    1 2x3 1 1 2x4 1 b fln 1 x g2 x2 1 2 3 2 3 4 MISCELLANEOUS PROBLEMS
for 1 < x @ 1
11.110. Prove that the series for Jp x converges (a) for all x, (b) absolutely and uniformly in any nite interval. 11.111. Prove that (a) d fJ x g J1 x ; dx 0 b d p fx Jp x g xp Jp 1 x ; dx c Jp 1 x 2p J x Jp 1 x . x p
11.112. Assuming that the result of Problem 11.111(c) holds for p 0; 1; 2; . . . ; prove that (a) J 1 x J1 x ; b J 2 x J2 x ; c J n x 1 n Jn x ; n 1; 2; 3; . . . : 11.113. Prove that e1=2x t 1=t
1 X p 1
Jp x tp .
[Hint: Write the left side as ext=2 e x=2t , expand and use Problem 11.112.]
CHAP. 11]
1 X n 1 zn n 1 1 X n 1
INFINITE SERIES
11.114. Prove that
n n 2 2
1 X n 1
is absolutely and uniformly convergent at all points within and on the circle jzj 1.
11.115. (a) If
an xn
bn xn for all x in the common interval of convergence jxj < R where R > 0, prove that (b) Use (a) to show that the Taylor expansion of a function exists, the
an bn for n 0; 1; 2; . . . . expansion is unique.
p 11.116. Suppose that lim n jun j L. Prove that un converges or diverges according as L < 1 or L > 1. If L 1 the test fails. 11.117. Prove that the radius of convergence of the series an xn can be determined by the following limits, when   a  1 1 they exist, and give examples: (a) lim  n ; b lim p ; c lim p : n!1an 1  n!1 n jan j n!1 n jan j 11.118. Use Problem 11.117 to nd the radius of convergence of the series in Problem 11.22. 11.119. (a) Prove that a necessary and su cient condition that the series un converge is that, given any  > 0, we can nd N > 0 depending on  such that jSp Sq j <  whenever p > N and q > N, where Sk u1 u2 uk . b Use a to prove that the series
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