Chapter 8.1, Problem 51E

To determine

The sequence is increasing, decreasing or not monotonic and bounded or not bounded.

Answer to Problem 51E

The sequence is monotone decreasing if n is an even number or the sequence is monotone increasing if n is an odd number and the sequence is not bounded.

Explanation of Solution

Given information: The sequence is $a_n=n{(-1)}^n$

Definitions:

Monotone increasing: A sequence ${\left\{x_n\right\}}_n$ is said to be monotone increasing, iff $\forall n,x_{n+1}\ge x_n.$

Monotone decreasing: A sequence ${\left\{x_n\right\}}_n$ is said to be monotone decreasing iff $\forall n,x_{n+1}\le x_n.$

Given sequence, $a_n=n{(-1)}^n$

In the given sequence if n is an even number then $a_n=n$

Now we find $a_{n+1}$

  $a_{n+1}=-n-1$

From the above definition, we have $a_{n+1}$ $-a_n$

  $\begin{array}{l}=-n-1-n\\ =-2n-1\\ =-(2n+1)\end{array}$

Thus, $-(2n+1)<0$

Hence, $a_{n+1}$ $<a_n$

From the above definition , the sequence is monotone decreasing if n is an even number.

In the given sequence if n is an odd number then $a_n=-n$

Now we find $a_{n+1}$

  $a_{n+1}=n+1$

From the above definition, we have $a_{n+1}$ $-a_n$

  $\begin{array}{l}=n+1+n\\ =2n+1\\ =(2n+1)\end{array}$

Thus, $(2n+1)>0$

Hence,

  $a_{n+1}$ $>a_n$

From the above definition , the sequence is monotone increasing if n is an odd number.

  $\text{ But, for any given }G>0,\exists n\text{ s}\text{.t}\text{. }\left|{(-1)}^{n}n\right|=n>G$

Hence, the given sequence is not bounded.