Answer
Converges
Work Step by Step
Given $$\sum_{n=1}^{\infty}\left(\frac{n}{5 n+2}\right)^{n}$$
By using the $n$th root test
\begin{align*}
\lim_{n\to\infty} \sqrt[n]{a_n}&=\lim_{n\to\infty} \sqrt[n]{\left(\frac{n}{5 n+2}\right)^{n}}\\
&=\lim_{n\to\infty} \left(\frac{n}{5 n+2}\right) \\
&=\frac{1}{5}<1
\end{align*}
Thus the given series converges.
