Chapter 8, Problem 7RE

To determine

To find: Whether the sequence is convergent or divergent. If it is convergent then find its limit.

Answer to Problem 7RE

The given sequence is convergent, and limit is $e^{12}$ .

Explanation of Solution

Given:

The series is $a_n=\left\{{\left(1+3/n\right)}^{4n}\right\}$ .

Calculation:

It is known that the sequence $S_n={\left(1+\frac{1}{n}\right)}^n$ is convergent and $\underset{n\to \infty }{\lim }{\left(1+\frac{1}{n}\right)}^n=e$ .

Thus, $\underset{n\to \infty }{\lim }{\left(1+\frac{3}{n}\right)}^n=e^3$ and $\underset{n\to \infty }{\lim }{\left(1+\frac{3}{n}\right)}^{4n}=e^{12}$

The limit of the series exists.

Therefore, the given sequence is convergent, and limit is $e^{12}$ .