Chapter 11.4, Problem 3E

To determine

To find the limit of the sequence

  $a_n=5^n$

Answer to Problem 3E

The limit of the given sequence does not exist.

Explanation of Solution

Given:

  $a_n=5^n$

Concept Used:

1. 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

  $a_n=5^n$

Find the value of

  $\underset{n\to \infty }{\lim }{a}_n=\underset{n\to \infty }{\lim }{5}^n$

Since, here

  $\begin{array}{c}\left|a_n\right|=\left|5\right|\\ =5\\ >1\forall n\end{array}$

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

  $\begin{array}{c}\underset{n\to \infty }{\lim }{a}_n=\underset{n\to \infty }{\lim }{5}^n\\ =\infty \end{array}$

Thus, limit of given sequence is $\infty$ , thus the limit of the given sequence does not exist.