Chapter B, Problem 11AYU

To determine

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

Answer to Problem 11AYU

Solution:

The given sequence converges. The limit is 0.

Explanation of Solution

Given:

$\left\{a_n\right\} = \left\{\frac{5n}{n^2 + 1}\right\}$

Calculation:

$\left\{a_n\right\} = \left\{\frac{5n}{n^2 + 1}\right\}$

$\left\{\frac{5n}{n^2 + 1}\right\} = \left\{\frac{n\left(5\right)}{n\left(n + \frac{1}{n}\right)}\right\} = \frac{5}{\left(n + \frac{1}{n}\right)}$

Hence the sequence converges. Therefore we will find the limit

$\underset{n\to \infty }{\lim }{a}_n=\underset{n\to \infty }{\text{lim}}\frac{5}{\left(n+\frac{1}{n}\right)}=0$