Chapter 11.4, Problem 2E

To determine

To find the limit of the sequence

  $a_n={\left(-\frac{1}{3}\right)}^n$

Answer to Problem 2E

0.

Explanation of Solution

Given:

  $a_n={\left(-\frac{1}{3}\right)}^n$

Concept Used:

Limit of a sequence $a_n$ is $\underset{n\to \infty }{\lim }{a}_n$ this tends to

 1.  0 if $\left|a_n\right|<1\forall n$

 2. $\infty$ if $\left|a_n\right|>1\forall n$

Calculation:

In order to find the limit of the sequence

  $a_n={\left(-\frac{1}{3}\right)}^n$

Find the value of

  $\underset{n\to \infty }{\lim }{a}_n=\underset{n\to \infty }{\lim }{\left(-\frac{1}{3}\right)}^n$

Since, here

  $\begin{array}{c}\left|a_n\right|=\left|-\frac{1}{3}\right|\\ =\frac{1}{3}\\ <1\forall n\end{array}$

Thus, the value of $\underset{n\to \infty }{\lim }{a}_n$ tends to 0 when n tends to infinity, i.e.

  $\begin{array}{c}\underset{n\to \infty }{\lim }{a}_n=\underset{n\to \infty }{\lim }{\left(-\frac{1}{3}\right)}^n\\ =0\end{array}$

Thus, limit of given sequence is 0.