Answer
Convergent
Work Step by Step
Given: $a_n=\dfrac{n}{5n+1}$
Here, we have
$\lim\limits_{n\to \infty}a_n=\lim\limits_{n\to \infty}\dfrac{n}{5n+1}$
This gives:
$\lim\limits_{n\to \infty}a_n=\lim\limits_{n\to \infty}\dfrac{n/n}{5n+1/n}$
or, $=\dfrac{\lim\limits_{n\to \infty}(1)}{\lim\limits_{n\to \infty} 5+\lim\limits_{n\to \infty}(1/n)}$
Thus, $a_n=\dfrac{1}{5}$
Hence, the sequence is convergent.