Page 313 - Schaum's Outline of Theory and Problems of Advanced Calculus

304 INFINITE SERIES [CHAP. 11

n 1
1
X
11.127. Prove that ð 1Þ converges uniformly for all x, but not absolutely.
n þ x 2
n¼1
1 1 1 1
ln 2
ffiffiffi
11.128. Prove that 1 þ þ ¼ p þ
4 7 10 3 3 3
n 1 n 1
1
X n
y ð 1Þ n
n!
11.129. If x ¼ ye ,prove that y ¼ x for 1=e < x @ 1=e.
n¼1

11.130. Prove that the equation e ¼ 1 has only one real root and show that it is given by
n 1 n 1 n
1
X n e
ð 1Þ
n!
¼ 1 þ
n¼1
x B 2 x 2 B 3 x 3
11.131. Let ¼ 1 þ B 1 x þ þ þ . (a) Show that the numbers B n ,called the Bernoulli numbers,
x
e 1 2! 3!
n
k
n
satisfy the recursion formula ðB þ 1Þ B ¼ 0 where B is formally replaced by B k after expanding.
(b)Using (a)orotherwise, determine B 1 ; ... ; B 6 .
Ans: 1 1 1 ; B 5 ¼ 0; B 6 ¼ .
1
2 6 30 42
ðbÞ B 1 ¼ ; B 2 ¼ ; B 3 ¼ 0; B 4 ¼
x x x
11.132. (a)Prove that coth 1 : ðbÞ Use Problem 11.127 and part (a)toshow that B 2kþ1 ¼ 0if
x
e 1 ¼ 2 2
k ¼ 1; 2; 3; .. . :
11.133. Derive the series expansions:
1 x x 3 B 2n ð2xÞ 2n
x 3 45 ð2nÞ!x
ðaÞ coth x ¼ þ þ þ þ
1 x x 3 2n
n B 2n ð2xÞ
x 3 45 ð2nÞ!x
ðbÞ cot x ¼ þ ð 1Þ þ
x 3 2x 5 2n 2n 1
n 1 2ð2 1ÞB 2n ð2xÞ
ðcÞ tan x ¼ x þ þ þ ð 1Þ þ
3 15 ð2nÞ!
1 x 7 3 n 1 2ð2 2n 1 1ÞB 2n x 2n 1
x 6 360 ð2nÞ!
ðdÞ csc x ¼ þ þ x þ ð 1Þ þ
[Hint: For (a)use Problem 11.132; for (b) replace x by ix in (a); for (c)use tan x ¼ cot x 2cot2x;for (d)use
csc x ¼ cot x þ tan x=2.]

1
Y 1
11.134. Prove that 1 þ converges.
n 3
n¼1

1
Y 1
11.135. Use the definition to prove that 1 þ diverges.
n
n¼1
1
Y
11.136. Prove that ð1 u n Þ, where 0 < u n < 1, converges if and only if u n converges.
n¼1

1
Y 1
1
11.137. (a)Prove that 1 converges to .(b) Evaluate the infinite product in (a)to2decimal places and
2
n 2
n¼2
compare with the true value.
11.138. Prove that the series 1 þ 0 1 þ 1 þ 0 1 þ 1 þ 0 1 þ is the C 1summable to zero.