Answer
Convergent
Work Step by Step
The Comparison Test states that the p-series $\sum_{n=1}^{\infty}\frac{1}{n^{p}}$ is convergent if $p\gt 1$ and divergent if $p\leq 1$.
$a_{n}=\frac{n}{n^{3}+1}\lt \frac{n}{n^{3}}=\frac{1}{n^{2}}$
Since, $\sum_{n=1}^{\infty}\frac{1}{n^{2}}$ is convergent, $\sum_{n=1}^{\infty}\frac{n}{n^{3}+1}$ is also convergent.
Hence, the series $\sum_{n=1}^{\infty}\frac{n}{n^{3}+1}$ is convergent.
