2.5 Monotone Sequences 89- Limit Comparison Test: Suppose {xn} and {Yn} are sequences of
positive real numbers such that Xn ---+ L , where 0 < L < oo. Prove that
Xn ---+ 00 ¢:? Yn ---+ 00. Yn - Geometric Series: Given t hat r > 1, find
(X)
L ark = lim (a+ ar + ar^2 + · · · + arn) and justify your answer. [See
n-+=
k=O
Exercise 2.3.6.]
1 6. Prove the relations (a) given in Table 2.2.17. Prove the relations (b) given in Table 2.2.- Prove the relations ( c) given in Table 2.2.
- Prove the relations (d) given in Table 2.2.
- Prove the relation (a) given in Table 2.3.
2 1. Prove the relation (b) given in Table 2.3.2.5 Monotone Sequences
One of the most powerful tools in the theory and application of sequences
is the notion of monotone increasing or monotone decreasing sequences. Such
sequences have special convergence behavior that make them especially useful.
Definition 2.5.1 A sequence {an} is said to be (see Figure 2.5)(a) monotone increasing if \in E N, an ::; an+ I; that is,(b) monotone decreasing if \in E N, an 2: an+I; that is ,(c) strictly increasing if\in E N,an < an+ 1 ; that is,
( d) strictly decreasing if \in E N, an > an+I; that is ,
ai > a2 > · · · > an > an+I > · · ·.