Answer
Diverges
Work Step by Step
Given $$ \sum_{n=2}^{\infty} \frac{n^3}{\sqrt{n^7+2n^2+1}}$$
Compare with the divergent series $\displaystyle \sum_{n=2}^{\infty}\frac{1}{n^{1/2}}$ ($p-$series , p<1) and by using the Limit Comparison Test, we get:
\begin{align*}
\lim_{n\to \infty} \frac{a_n}{b_n}&=\lim_{n\to \infty} \frac{n^{7/2}}{\sqrt{n^7+2n^2+1}}\\
&=1
\end{align*}
Then $\displaystyle \sum_{n=2}^{\infty} \frac{n^3}{\sqrt{n^7+2n^2+1}}$ also diverges
