Chapter 8, Problem 3RE

To determine

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

Answer to Problem 3RE

The given sequence is divergent.

Explanation of Solution

Given:

The series is $a_n=\frac{n^3}{1+n^2}$ .

Calculation:

To check the convergence of the series find the limit $n\to \infty$ , and if the limit exists then $a_n$ is convergent otherwise divergent.

Find the limit of the series $a_n=\frac{n^3}{1+n^2}$ at $n\to \infty$ .

  $\begin{array}{c}\underset{n\to \infty }{\lim }{a}_n=\underset{n\to \infty }{\lim }\left(\frac{n^3}{1+n^2}\right)\\ =\underset{n\to \infty }{\lim }\left(\frac{n}{\frac{1}{n^2}+1}\right)\\ =\frac{\infty }{0+1}\\ =\infty \end{array}$

The limit of the series does not exist.

Therefore, the given sequence is divergent.