Page 46 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 2] SEQUENCES 37

Ans. (a) 1 ; 1; 0; 0 ðbÞ 1; 1; 1; 1 (d) none, 1; þ1; 1
3 ðcÞ none, none, þ1, 1
2.51. Prove that a bounded sequence fu n g is convergent if and only if lim u n ¼ lim u n .

INFINITE SERIES
1
X n
2.52. Find the sum of the series 2 . Ans. 2
3
n¼1
1
X n 1 n
2.53. Evaluate =5 . Ans. 1
6
ð 1Þ
n¼1
1 1 1 1 X 1 1 1 1
1
2.54. Prove that ¼ 1. Hint:
1 2 þ 2 3 þ 3 4 þ 4 5 þ ¼ ¼ n n þ 1
n¼1 nðn þ 1Þ nðn þ 1Þ
2.55. Prove that multiplication of each term of an infinite series by a constant (not zero) does not affect the
convergence or divergence.

1 1 1 1 1 1
2.56. Prove that the series 1 þ þ þ þ þ diverges. Hint: Let S n ¼ 1 þ þ þ þ . Then prove
2 3 n 2 3 n
1
that jS 2n S n j > ,giving a contradiction with Cauchy’s convergence criterion.
2
MISCELLANEOUS PROBLEMS
2.57. If a n @ u n @ b n for all n > N, and lim a n ¼ lim b n ¼ l,prove that lim u n ¼ l.
n!1 n!1 n!1
2.58. If lim a n ¼ lim b n ¼ 0, and is independent of n,prove that lim ða n cos n þ b n sin n Þ¼ 0. Is the result
n!1 n!1 n!1
true when depends on n?
n
1
1
2.59. Let u n ¼ f1 þð 1Þ g, n ¼ 1; 2; 3; ... . If S n ¼ u 1 þ u 2 þ þ u n ,prove that lim S n =n ¼ .
2 2
n!1
2.60. Prove that (a) lim n 1=n ¼ 1, (b) lim ða þ nÞ p=n ¼ 1 where a and p are constants.
n!1 n!1
2.61. If lim ju nþ1 =u n j¼jaj < 1, prove that lim u n ¼ 0.
n!1 n!1
p n
2.62. If jaj < 1, prove that lim n a ¼ 0 where the constant p > 0.
n!1
n
2 n!
2.63. Prove that lim ¼ 0.
n n
2.64. Prove that lim n sin 1=n ¼ 1. Hint: Let the central angle, ,ofa circle be measured in radians. Geome-
n!1
trically illustrate that sin tan ,0 .
Let ¼ 1=n. Observe that since n is restricted to positive integers, the angle is restricted to the first
quadrant.
1
5Þ.
2.65. If fu n g is the Fibonacci sequence (Problem 2.33), prove that lim u nþ1 =u n ¼ ð1 þ p ffiffiffi
2
n!1
2.66. Prove that the sequence u n ¼ð1 þ 1=nÞ nþ1 , n ¼ 1; 2; 3; .. . is a monotonic decreasing sequence whose limit
is e. [Hint: Show that u n =u n 1 @ 1:Š
2.67. If a n A b n for all n > N and lim a n ¼ A, lim b n ¼ B,prove that A A B.
n!1 n!1
2.68. If ju n j @ jv n j and lim v n ¼ 0, prove that lim u n ¼ 0.
n!1 n!1

1 1 1 1
2.69. Prove that lim ¼ 0.
3
n!1 n 1 þ þ þ þ n
2