Answer
Diverges
Work Step by Step
Given $$ \sum_{n=2}^{\infty} \frac{n}{\sqrt{n^3+1}}$$ Compare with the divergent series $\displaystyle \sum_{n=2}^{\infty}\frac{1}{n^{1/2}}$
Use the Limit Comparison Test \begin{align*} \lim_{n\to \infty} \frac{a_n}{b_n}&=\lim_{n\to \infty} \frac{n^{1/2}}{\sqrt{n^3+1}}\\ &=0 \end{align*} Then $\displaystyle \sum_{n=2}^{\infty}\frac{n}{\sqrt{n^3+1}}$ also diverges.
