Chapter 13, Problem 37RE

To determine

To determine whether the sequence is convergent or divergent and then explain if divergent or find limit if convergent

Answer to Problem 37RE

The sequence is divergent.

Explanation of Solution

Given information:

Consider the given information,

  $a_n=\frac{n}{5n+1}$

CALCULATION:

To find the sequence as,

  $\begin{array}{l}{a}_n=\frac{n}{5n+1}\\ \underset{n\to \infty }{\lim }{a}_n=\underset{n\to \infty }{\lim }\frac{n}{5n+1}\\ =\underset{n\to \infty }{\lim }\frac{n/n}{\frac{5n}{n}+\frac{1}{n}}\left(\text{dividing numerator and denominator by n}\right)\\ =\underset{n\to \infty }{\lim }\frac{1}{5+\frac{1}{n}}=\frac{1}{5+0}=\frac{1}{5}\ne 0\ne \infty \end{array}$

Therefore, as the limit $\ne 0,$ so the sequence is divergent.