Page 308 - Schaum's Outline of Theory and Problems of Advanced Calculus
P. 308
CHAP. 11] INFINITE SERIES 299
11.65. Use the comparison test to prove that
1
1 1 1 X 2
X 1 X tan n n
@ diverges, converges.
ðaÞ p converges if p > 1 and diverges if p @ 1; ðbÞ ðcÞ n
n n 2
n¼1 n¼1 n¼1
11.66. Establish the results (b) and (c)ofthe quotient test, Page 267.
11.67. Test for convergence:
2
1 1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1
X X X 3 þ sin n X
; n tan ð1=n Þ; ; n sin ð1=nÞ:
ðln nÞ 1 3 2
n
ðaÞ 2 ðbÞ ðcÞ n ðdÞ
n¼1 n¼1 n¼1 nð1 þ e Þ n¼1
Ans. (a)conv., (b)div., (c)div., (d)div.
11.68. If u n converges, where u n A 0for n > N, and if lim nu n exists, prove that lim nu n ¼ 0.
n!1 n!1
1
1
X
11.69. (a)Test for convergence . (b) Does your answer to (a)contradict the statement about the p
n 1þ1=n
n¼1
p
series made on Page 266 that 1=n converges for p > 1?
Ans. (a)div.
INTEGRAL TEST
p
1 2 1 1 X n ffiffi 1
1
X n X 1 X n e X ln n
11.70. Test for convergence: (a) 3 ; ðbÞ 3 ; ðcÞ n ; ðdÞ p ffiffiffi ðeÞ ;
2n 1 2 n n
n¼1 n¼2 nðln nÞ n¼1 n¼1 n¼2
1 lnðln nÞ
X 2
:
ð f Þ
n ln n
n¼10
Ans: ðaÞ div., ðbÞ conv., ðcÞ conv., ðdÞ conv., ðeÞ div., ð f Þ div.
1
X 1
11.71. Prove that p , where p is a constant, (a)converges if p > 1 and (b)diverges if p @ 1.
n¼2 nðln nÞ
9 X 1 5
1
11.72. Prove that < < .
8 n 3 4
n¼1
1
X tan n
e
1
11.73. Investigate the convergence of :
2
n þ 1
n¼1
Ans: conv.
2 3=2
2 3=2
2
ffiffiffi
11.74. (a)Prove that n 1 @ p ffiffiffi p ffiffiffi p ffiffiffi p n @ n þ n 1=2 .
3 þ 3 1 þ 2 þ 3 þ þ 3 3
p ffiffiffi p ffiffiffi p ffiffiffi p ffiffiffiffiffiffiffiffi
(b)Use (a)to estimate the value of 1 þ 2 þ 3 þ þ 100,giving the maximum error.
p ffiffiffiffiffi p ffiffiffiffiffi p ffiffiffiffiffiffiffiffi
(c) Show how the accuracy in (b)can be improved by estimating, for example, 10 þ 11 þ þ 100
p ffiffiffi p ffiffiffi p ffiffiffi
and adding on the value of 1 þ 2 þ þ 9 computed to some desired degree of accuracy.
Ans: ðbÞ 671:5 4:5
ALTERNATING SERIES
nþ1 n nþ1
1 1 1
X X X n
11.75. Test for convergence: (a) ð 1Þ ; ðbÞ ð 1Þ ; ðcÞ ð 1Þ ;
2
2 n n þ 2n þ 2 3n 1
n¼1 n¼1 n¼1
1 1 n p ffiffiffi
X n 1 1 X ð 1Þ n
ð 1Þ sin ; :
ðdÞ ðeÞ
n ln n
n¼1 n¼2
Ans. (a)conv., (b)conv., (c)div., (d)conv., (e)div.
