Answer
Converges
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{n^3}{n^5 + 4n+1}$$
We compare the given series with the series $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^2 }$, which is a convergent series ( $p-$ series with $p=2$) and for $n>1$
$$ \frac{1}{n2^n}\leq\frac{1}{2^n} $$
Then $\displaystyle\sum_{n=1}^{\infty} \frac{n^3}{n^5 + 4n+1}$ also converges.
