88 Chapter 2 11 Sequences
- Use Definition 2.4.1 to prove each of the limit statements in Exercise 1.
- Prove each of the following limit statements, using the theorems of this
and previous sections:
( ) 1 3n + sn (b) l" n
(^3) + sin(3n)
a n->oo ~ 7n =+oo n->oo rm n - 1 =+oo
. n!
(c) hm 100 = +oo
n-+oo n
4. Prove Theorem 2.4.4.
5. Prove Theorem 2.4.7 (b).
6. Prove Theorem 2.4.9 (b).
7. Prove Theorem 2.4.9 (c).
8. Prove Theorem 2.4.9 (d).
9. In each of the following parts (a)- (d), find sequences {an} and {bn} such
that an , +oo, bn , +oo, and the given condition holds:
(a) an - bn , 0 (b) an - bn , +oo
(c) an - bn , -oo (d) an - bn , L =f. 0
10. In each of the following parts (a)-( d), find sequences {an} and { bn} such
that an , +oo, bn , 0, and the given condition holds:
(a) anbn , 0 (b) anbn , +oo
(c) anbn , -oo (d) anbn , L =f. 0
11. Prove directly from the definitions that if an , +oo and {bn} is a se-
quence of positive terms bounded away from 0, then anbn , +oo.
12. Let c E JR and p E N be fixed. Prove that
n { 0 if lei ~ 1;
1. c
n->oo im -nP = +oo if c > l;
does not exist (finite or infinite) if c < -1.
[See Exercise 2.3.7.]- Prove that if {xn} is a sequence of positive numbers such that
(^1) im. --Xn+l = L > 1, t h en , Xn ___, +oo.
n-+oo Xn