Answer
Converges
Work Step by Step
Here, we have the given series as $\Sigma_{n=1}^{\infty} \dfrac{1}{ \sqrt {n^3+2n}}$
Let us consider that $a_n=\Sigma_{n=1}^{\infty} \dfrac{1}{ \sqrt {n^3+2n}}$ and $b_n=\Sigma_{n=1}^{\infty} \dfrac{1}{ \sqrt{n^3}}$
We can see that $a_n \leq b_n$ and $b_n$ shows a p-series with $p=\dfrac{3}{2}$ and $p=\dfrac{3}{2} \gt 1$
This implies that the series $b_n$ converges by the p-series test.
Hence, the given series also converges by the direct comparison test.
