Answer
Converges
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{3} + 2n-1}}$$
We compare the given series with the series $\displaystyle\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{3}} }$, which is a convergent series ( $p-$series with $p=3/2$) and for $n\geq1$
$$ \frac{1}{\sqrt{n^{3} + 2n-1}}\leq \frac{1}{\sqrt{n^{3} }} $$
Then $\displaystyle\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{3} + 2n-1}}$ also converges.
