Page 45 - Schaum's Outline of Theory and Problems of Advanced Calculus

36 SEQUENCES [CHAP. 2

4
4 2n 2 p ffiffi n þ 1 sin n
ðaÞ lim ; ðbÞ lim 2 1= n ¼ 1; ðcÞ lim ¼1; ðdÞ lim ¼ 0:
n!1 3n þ 2 ¼ 3 n!1 n!1 n 2 n!1 n
2.35. Find the least positive integer N such that jð3n þ 2Þ=ðn 1Þ 3j < for all n > N if (a) ¼ :01,
(b) ¼ :001, (c) ¼ :0001.
Ans.(a) 502, (b) 5002, (c) 50,002
1
2.36. Using the definition of limit, prove that lim ð2n 1Þ=ð3n þ 4Þ cannot be .
2
n!1
n
2.37. Prove that lim ð 1Þ n does not exist.
n!1
2.38. Prove that if lim ju n j¼ 0then lim u n ¼ 0. Is the converse true?
n!1 n!1
p
2
2
2.39. If lim u n ¼ l,prove that (a) lim cu n ¼ cl where c is any constant, (b) lim u n ¼ l , (c) lim u n ¼ l p
n!1 n!1 ffiffi n!1 n!1
p
ffiffiffiffiffi
where p is a positive integer, (d) lim p u n ¼ l; l A 0.
n!1
2.40. Give a direct proof that lim a n =b n ¼ A=B if lim a n ¼ A and lim b n ¼ B 6¼ 0.
n!1 n!1 n!1
2 1=n
3 n
2.41. Prove that (a) lim 3 1=n ¼ 1, (b) lim ¼ 1, (c) lim ¼ 0.
3 4
n!1 n!1 n!1
n
2.42. If r > 1, prove that lim r ¼1,carefully explaining the significance of this statement.
n!1
n
2.43. If jrj > 1, prove that lim r does not exist.
n!1
2.44. Evaluate each of the following, using theorems on limits:
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
4 2n 3n 2 3n 5n þ 4 p ffiffiffiffiffiffiffiffiffiffiffiffiffi
lim ðcÞ lim 2
2n þ n 2n 7
ðaÞ 2 ðeÞ lim ð n þ n nÞ
n!1 n!1 n!1
r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffi
p ffiffiffi p n 2n
3 ð3 nÞð n þ 2Þ 4 10 3 10 n n 1=n
lim ðdÞ lim
8n 4 n!1 3 10 þ 2 10
ðbÞ n 1 2n 1 ð f Þ lim ð2 þ 3 Þ
n!1 n!1
ffiffiffi
Ans: ðaÞ 3=2; ðbÞ 1=2; ðcÞ 3=2; ðdÞ 15; ðeÞ 1=2; ð f Þ 3
p
BOUNDED MONOTONIC SEQUENCES
2.45. Prove that the sequence with nth term u n ¼ p ffiffiffi n=ðn þ 1Þ (a)is monotonic decreasing, (b)is bounded below,
(c)is bounded above, (d) has a limit.
1 1 1 1
2.46. If u n ¼ þ þ þ þ ,prove that lim u n exists and lies between 0 and 1.
1 þ n 2 þ n 3 þ n n þ n n!1
ffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffi
p
2.47. 1 5Þ.
2
u n þ 1, u 1 ¼ 1, prove that lim u n ¼ ð1 þ
If u nþ1 ¼
n!1
1
2.48. If u nþ1 ¼ ðu n þ p=u n Þ where p > 0 and u 1 > 0, prove that lim u n ¼ p ffiffiffi p. Show how this can be used to
determine 2 ffiffiffip 2. n!1
2.49. If u n is monotonic increasing (or monotonic decreasing), prove that S n =n, where S n ¼ u 1 þ u 2 þ þ u n ,is
also monotonic increasing (or monotonic decreasing).
LEAST UPPER BOUND, GREATEST LOWER BOUND, LIMIT SUPERIOR, LIMIT INFERIOR
2.50. Find the l.u.b., g.l.b., lim sup (lim), lim inf (lim) for each sequence:
n
1
1 1
(a) 1; ; ; ; .. . ; ð 1Þ =ð2n 1Þ; .. . n 1 ð2n 1Þ; ...
3 5 7 ðcÞ 1; 3; 5; 7; ... ; ð 1Þ
(b) 2 3 4 5 nþ1 ðn þ 1Þ=ðn þ 2Þ; .. . ðdÞ 1; 4; 1; 16; 1; 36; .. . ; n 1þð 1Þn ; ...
6
4 5
3 ; ; ; ; .. . ; ð 1Þ