Answer
converges
Work Step by Step
Given $$\sum_{n=2}^{\infty} \frac{n}{\sqrt{n^{5}+5}}$$
Compare with $\sum\frac{1}{n^{3/2}}$, a convergent series ($p>1$):
\begin{align*}
\lim_{n\to\infty } \frac{a_n}{b_n}&=\lim_{n\to\infty } \frac{n^{5/2}}{\sqrt{n^{5}+5}}\\
&=\lim_{n\to\infty } \frac{n^{5/2} /n^{5/2}}{\sqrt{1+5/n^{5}}}\\
&=1
\end{align*}
Hence, $\sum_{n=2}^{\infty} \frac{n}{\sqrt{n^{5}+5}}$ also converges.
