Page 312 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 11] INFINITE SERIES 303

1 n
X ðn þ 1Þz
11.114. Prove that is absolutely and uniformly convergent at all points within and on the circle jzj¼ 1.
2
n¼1 nðn þ 2Þ
1 1
X X
n
n
11.115. (a)If a n x ¼ b n x for all x in the common interval of convergence jxj < R where R > 0, prove that
n¼1 n¼1
a n ¼ b n for n ¼ 0; 1; 2; ... . (b)Use (a)toshow that the Taylor expansion of a function exists, the
expansion is unique.
ffiffiffiffiffiffiffiffi
n
11.116. Suppose that lim ju n j ¼ L.Prove that u n converges or diverges according as L < 1or L > 1. If L ¼ 1
p
the test fails.
n
11.117. Prove that the radius of convergence of the series a n x can be determined by the following limits, when
1 1

they exist, and give examples: (a) lim a n ; ffiffiffiffiffiffiffiffi ; ffiffiffiffiffiffiffiffi :
n
n
ðbÞ lim p ðcÞ lim p
n!1 a nþ1 n!1 n!1
ja n j ja n j
11.118. Use Problem 11.117 to find the radius of convergence of the series in Problem 11.22.
11.119. (a)Prove that a necessary and sufficient condition that the series u n converge is that, given any > 0, we
can find N > 0depending on such that jS p S q j < whenever p > N and q > N, where
S k ¼ u 1 þ u 2 þ þ u k .
n
1
X
ðbÞ Use ðaÞ to prove that the series converges.
ðn þ 1Þ3 n
n¼1
1 1
X
ðcÞ How could you use ðaÞ to prove that the series diverges?
n
n¼1
[Hint: Use the Cauchy convergence criterion, Page 25.]
11.120. Prove that the hypergeometric series (Page 276) (a)is absolutely convergent for jxj < 1, (b)is divergent
for jxj > 1, (c)is absolutely divergent for jxj¼ 1if a þ b c < 0; ðdÞ satisfies the differential equation
xð1 xÞy þfc ða þ b þ 1Þxgy aby ¼ 0.
0
00
11.121. If Fða; b; c; xÞ is the hypergeometric function defined by the series on Page 276, prove that
p
2
1 1 3
(a) Fð p; 1; 1; xÞ¼ ð1 þ xÞ ; ðbÞ xFð1; 1; 2; xÞ¼ lnð1 þ xÞ; ðcÞ Fð ; ; ; x Þ¼ ðsin 1 xÞ=x.
2 2 2
x 3 x 5
þ .
1 3 1 3 5
11.122. Find the sum of the series SðxÞ¼ x þ þ
[Hint: Show that S ðxÞ 1 þ xSðxÞ and solve.]
0
ð x
2 2
Ans: e x =2 e x =2 dx
0
11.123. Prove that
1 1 1 p ffiffiffi 1 1 1 1
1 3 1 3 5 1 3 5 7 2 3 2 2! 5 2 3! 7 2 4! 9
1 þ þ þ þ ¼ e 1 þ 2 3 þ 4
11.124. Establish the Dirichlet test on Page 270.
sin nx
1
X
11.125. Prove that is uniformly convergent in any interval which does not include 0; ; 2 ; ... .
n
n¼1
[Hint: use the Dirichlet test, Page 270, and Problem 1.94, Chapter 1.]
11.126. Establish the results on Page 275 concerning the binomial series.
[Hint: Examine the Lagrange and Cauchy forms of the remainder in Taylor’s theorem.]