Page 307 - Schaum's Outline of Theory and Problems of Advanced Calculus

298 INFINITE SERIES [CHAP. 11

2 2 2
x þ 3y 2 ¼ 10 4ðx 1Þþ 4ð y þ 2Þ 2ðx 1Þ þ 2ðx 1Þð y þ 2Þþ ðx 1Þ ð y þ 2Þ
(Check this algebraically.)

x þ y x þ y 2
11.55. Prove that ln ¼ ; 0 < < 1; x > 0; y > 0. Hint: Use the Taylor formula
2 2 þ ðx þ y 2Þ
with the linear term as the remainder.

to second-degree terms.
11.56. Expand f ðx; yÞ¼ sin xy in powers of x 1 and y
2
1 2 2 2

8 2 2 2
1 ðx 1Þ ðx 1Þ y y

Supplementary Problems

CONVERGENCE AND DIVERGENCE OF SERIES OF CONSTANTS
1 1 1 X 1
1
11.57. (a)Prove that the series converges and (b) find its sum.
3 7 þ 7 11 þ 11 15 þ ¼
n¼1 ð4n 1Þð4n þ 3Þ
Ans. (b)1/12
11.58. Prove that the convergence or divergence of a series is not affected by (a) multiplying each term by the
same non-zero constant, (b) removing (or adding) a finite number of terms.

11.59. If u n and v n converge to A and B, respectively, prove that ðu n þ v n Þ converges to A þ B.

3
3 2
3 n
3 3
11.60. Prove that the series þð Þ þð Þ þ ¼ ð Þ diverges.
2
2
2
2
11.61. Find the fallacy: Let S ¼ 1 1 þ 1 1 þ 1 1 þ . Then S ¼ 1 ð1 1Þ ð1 1Þ ¼ 1 and
S ¼ð1 1Þþð1 1Þþð1 1Þþ ¼ 0. Hence, 1 ¼ 0.
COMPARISON TEST AND QUOTIENT TEST
11.62. Test for convergence:
1 1 1 1 n 1
X 1 X n X n þ 2 X 3 X 1
ðaÞ 2 ; ðbÞ 2 ; ðcÞ p ffiffiffiffiffiffiffiffiffiffiffi ; ðdÞ n ; ðeÞ ;
n þ 1 4n 3 ðn þ 1Þ n þ 3 n 5 5n 3
n¼1 n¼1 n¼1 n¼1 n¼1
1
X 2n 1
ð3n þ 2Þn :
ð f Þ 4=3
n¼1
Ans: ðaÞ conv., ðbÞ div., ðcÞ div., ðdÞ conv., ðeÞ div., ð f Þ conv.
1 2 1 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X 4n þ 5n 2 X n ln n
11.63. Investigate the convergence of (a) ; . Ans. (a)conv., (b)div.
2
2 3=2 ðbÞ n þ 10n 3
n¼1 nðn þ 1Þ n¼1
11.64. Establish the comparison test for divergence (see Page 267).