Page 44 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 2] SEQUENCES 35

2 n nðn 1Þ 2 nðn 1Þðn 2Þ 3 n
u n þ þ u n
2! 3!
1 þ n þ n ¼ð1 þ u n Þ ¼ 1 þ nu n þ u n þ
2
2 nðn 1Þðn 2Þ 3 3 6ðn þ nÞ
3! nðn 1Þðn 2Þ
Then 1 þ n þ n > 1 þ u n or 0 < u n < :
3
2 1=n
Hence, lim u n ¼ 0 and lim u n ¼ 0: Thus lim ð1 þ n þ n Þ ¼ lim ð1 þ u n Þ¼ 1:
n!1 n!1 n!1 n!1
a n
2.30. Prove that lim ¼ 0 for all constants a.
n!1 n!
n
The result follows if we can prove that lim jaj ¼ 0 (see Problem 2.38). We can assume a 6¼ 0.
n n!1 n!
u n
jaj jaj
. Then .If n is large enough, say, n > 2jaj, and if we call N ¼½2jajþ 1Š, i.e., the
n! u n 1 n
Let u n ¼ ¼
greatest integer @ 2jajþ 1, then
u Nþ1 1 u Nþ2 1 u n 1
< ; < ; .. . ; <
u N 2 u Nþ1 2 u n 1 2
1 n N
1 n N
u n
Multiplying these inequalities yields < 2 or u n < 2 u N :
u N
n N
1
Since lim ¼ 0(using Problem 2.7), it follows that lim u n ¼ 0.
n!1 2 n!1
Supplementary Problems
SEQUENCES
2.31. Write the first four terms of each of the following sequences:
( ) ( ) ( )
p ffiffiffi nþ1 n 1 n 2n 1
n ð 1Þ ð2xÞ ð 1Þ x cos nx
; ; ; ; :
ðaÞ ðbÞ ðcÞ 5 ðdÞ ðeÞ 2 2
n þ 1 n! x þ n
ð2n 1Þ 1 3 5 ð2n 1Þ
p ffiffiffi p ffiffiffi p ffiffiffi p ffiffiffi 2 3
1 2 3 4 1 2x 4x 8x cos x cos 2x cos 3x cos 4x
Ans: ðaÞ ; ; ; ðcÞ ; ; ; ðeÞ ; ; ;
2
2
2
2
2 3 4 5 1 5 3 5 5 5 7 5 x þ 1 2 x þ 2 2 x þ 3 2 x þ 4 2
1 1 1 1 x x 3 x 5 x 7
; ; ; ;
ðbÞ ; ; ðdÞ
1! 2! 3! 4! 1 1 3 1 3 5 1 3 5 7
2.32. Find a possible nth term for the sequences whose first 5 terms are indicated and find the 6th term:
1 3 5 7 9
; ; ; ; ; .. . ðbÞ 1; 0; 1; 0; 1; ... 2 ; 0; ; 0; ; ...
3
4
5 8 11 14 17
ðaÞ ðcÞ 3 4 5
n n n
Ans: ðaÞ ð 1Þ ð2n 1Þ ðbÞ 1 ð 1Þ ðcÞ ðn þ 3Þ 1 ð 1Þ
2 2

ð3n þ 2Þ ðn þ 5Þ
2.33. The Fibonacci sequence is the sequence fu n g where u nþ2 ¼ u nþ1 þ u n and u 1 ¼ 1, u 2 ¼ 1. (a)Find the first 6
ffiffiffi
n
n
ffiffiffi
p
1
terms of the sequence. (b) Show that the nth term is given by u n ¼ða b Þ= 5, where a ¼ ð1 þ p 5Þ,
2
1 p ffiffiffi 5Þ.
2
b ¼ ð1
Ans. (a)1; 1; 2; 3; 5; 8
LIMITS OF SEQUENCES
2.34. Using the definition of limit, prove that: