Page 309 - Schaum's Outline of Theory and Problems of Advanced Calculus

300 INFINITE SERIES [CHAP. 11

n
1
X
11.76. (a) What is the largest absolute error made in approximating the sum of the series ð 1Þ by the sum
n
of the first 5 terms? n¼1 2 ðn þ 1Þ
Ans. 1/192
(b) What is the least number of terms which must be taken in order that 3 decimal place accuracy will
result?
Ans. 8 terms
1 1 1 4 1 1 1
11.77. (a)Prove that S ¼ þ þ þ ¼ þ .
1 3 2 3 3 3 3 1 3 2 3 3 3
(b) How many terms of the series on the right are needed in order to calculate S to six decimal place
accuracy?
Ans. (b)at least 100 terms

ABSOLUTE AND CONDITIONAL CONVERGENCE
11.78. Test for absolute or conditional convergence:
n 1 n n 1
1 1 1
X X X 1
ð 1Þ ð 1Þ ð 1Þ
ðaÞ 2 ðcÞ ðeÞ sin p ffiffiffi
n þ 1 n ln n 2n 1 n
n¼1 n¼2 n¼1
n 1 n 3 n 1 3
1 1 1
X n X ð 1Þ n X n
ð 1Þ ð 1Þ
n þ 1 2 2 1
ðbÞ 2 ðdÞ 4=3 ð f Þ n
n¼1 n¼1 ðn þ 1Þ n¼1
Ans.(a) abs. conv., (b)cond. conv., (c)cond. conv., (d)div., (e) abs. conv., ( f ) abs. conv.
cos n a
1
X
11.79. Prove that converges absolutely for all real x and a.
2
x þ n 2
n¼1
1
1
1
1
1
1
1
1
1
1
1
11.80. If 1 þ þ converges to S,prove that the rearranged series 1 þ þ þ þ þ 11 þ
3
7
5
2
4
3
9
6
2
4
3
¼ S. Explain.
2
1
1
1
[Hint: Take 1/2 of the first series and write it as 0 þ þ 0 þ 0 þ þ ;then add term by term to the first
2 4 6
series. Note that S ¼ ln 2, as shown in Problem 11.100.]
11.81. Prove that the terms of an absolutely convergent series can always be rearranged without altering the sum.
RATIO TEST
11.82. Test for convergence:
1 n 1 2n X n 1 n 3n 1 p ffiffiffi n
1
X ð 1Þ n X 10 3 X ð 1Þ 2 X ð 5 1Þ
ðaÞ n ; ðbÞ ; ðcÞ 3 ; ðdÞ 2n ; ðeÞ 2 :
ðn þ 1Þe ð2n 1Þ! n 3 n þ 1
n¼1 n¼1 n¼1 n¼1 n¼1
Ans.(a)conv. (abs.), (b)conv., (c)div., (d)conv. (abs.), (e)div.
11.83. Show that the ratio test cannot be used to establish the conditional convergence of a series.
1 !
X n! n
11.84. Prove that (a) converges and (b) lim ¼ 0.
n n n!1 n n
n¼1
MISCELLANEOUS TESTS
11.85. Establish the validity of the nth root test on Page 268.
11.86. Apply the nth root test to work Problems 11.82ðaÞ,(c), (d), and (e).
2 2
1 5
2 4
1 3
2 6
1
11.87. Prove that þð Þ þð Þ þð Þ þð Þ þð Þ þ converges.
3 3 3 3 3 3