Chapter B, Problem 8AYU

In Problems 5-28, determine whether each sequence converges or diverges. If it converges, find its limit.

$\left\{b_n\right\}=\left\{\frac{n^2}{n+3}\right\}$

To determine

To find: Determine whether each sequence converges or diverges. If it converges, find its limit.

Answer to Problem 8AYU

Solution:

The given sequence diverges.

Explanation of Solution

Given:

$\left\{b_n\right\} = \left\{\frac{n^2}{n + 3}\right\}$

Calculation:

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

Hence the sequence Diverges.