Page 310 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 11] INFINITE SERIES 301

1 1 4 1 4 7 2 2 5 2 5 8
11.88. Test for convergence: (a) þ þ þ , (b) þ þ þ .
3 3 6 3 6 9 9 9 12 9 12 15
Ans.(a)div., (b)conv.
11.89. If a; b, and d are positive numbers and b > a,prove that
a aða þ dÞ aða þ dÞða þ 2dÞ
b þ bðb þ dÞ þ bðb þ dÞðb þ 2dÞ þ
converges if b a > d, and diverges if b a @ d.

SERIES OF FUNCTIONS
11.90. Find the domain of convergence of the series:
1 n 1 n n 1 1 n 1 nx
X x X ð 1Þ ðx 1Þ X 1 X 2 1 x X e
ðaÞ 3 ; ðbÞ n ; ðcÞ 2 n ; ðdÞ n 1 þ x ; ðeÞ 2
n¼1 n n¼1 2 ð3n 1Þ n¼1 nð1 þ x Þ n¼1 n¼1 n n þ 1
Ans: ðaÞ 1 @ x @ 1; ðbÞ 1 < x @ 3; ðcÞ all x 6¼ 0; ðdÞ x > 0; ðeÞ x @ 0
1
X
n
11.91. Prove that 1 3 5 ð2n 1Þ x converges for 1 @ x < 1.
n¼1 2 4 6 ð2nÞ
UNIFORM CONVERGENCE
11.92. By use of the definition, investigate the uniform convergence of the series
x
1
X
n¼1 ½1 þðn 1ÞxŠ½1 þ nxŠ

1
:
1 þ nx
Hint: Resolve the nth term into partial fractions and show that the nth partial sum is S n ðxÞ¼ 1
Ans. Not uniformly convergent in any interval which includes x ¼ 0; uniformly convergent in any other
interval.
11.93. Work Problem 11.30 directly by first obtaining S n ðxÞ.
11.94. Investigate by any method the convergence and uniform convergence of the series:
2
x
X n X sin nx X x
1
1
1
; ; ; x A 0:
3 2 1
ðaÞ ðbÞ n ðcÞ n
n¼1 n¼1 n¼1 ð1 þ xÞ
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.
1
X sin nx
11.95. If FðxÞ¼ ,prove that:
n 3
n¼1 X cos nx
1
(a) FðxÞ is continuous for all x, (b) lim FðxÞ¼ 0; ðcÞ F ðxÞ¼ is continous everywhere.
0
x!0 n 2
n¼1

ð cos 2x cos 4x cos 6x
11.96. Prove that þ þ þ dx ¼ 0.
0 1 3 3 5 5 7
1
X sin nx
11.97. Prove that FðxÞ¼ has derivatives of all orders for any real x.
sinh n
n¼1
1
11.98. Examine the sequence u n ðxÞ¼ ; n ¼ 1; 2; 3; .. . ; for uniform convergence.
1 þ x 2n
ð 1 dx
1
11.99. Prove that lim n ¼ 1 e .
n!1
0 ð1 þ x=nÞ