   Chapter 11.1, Problem 75E

Chapter
Section
Textbook Problem

# Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?75. an = n(−1)n

To determine

Whether the sequence is increasing, decreasing, not monotonic and bounded or not.

Explanation

Definition used:

(1) The sequence {an} is increasing if an<an+1 for all n1 . That is, a1<a2<a3<... .

(2) The sequence {an} is decreasing and an>an+1 for all n1 . That is, a1>a2>a3>... .

(3) If the sequence is either increasing or decreasing, then the sequence is called monotonic; otherwise it is not monotonic.

(4) If {an} is the sequence with manM for m,M , then the sequence is bounded.

Theorem used:

If limn|an|=0 , then limnan=0 . (1)

Given:

The sequence is an=n(1)n (2)

Calculation:

Check whether the sequence is increasing, decreasing or not monotonic.

The first five terms of the sequence is as follows:

{n(1)n}={1(1)1,2(1)2,3(1)3,4(1)4,5(1)5...}   ={1(1),2(1),3(1),4(1),5(1)

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