Answer
Conditionally convergent.
Work Step by Step
Given $$
\sum_{n=1}^{\infty}(-1)^{n} \frac{n}{\sqrt{n^{3}+2}}
$$
Use the Limit Comparison Test with $a_n =\dfrac{n}{\sqrt{n^{3}+2}}$ and $b_n=\dfrac{1}{\sqrt{n }}$
\begin{align*}
\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}&=\lim _{n \rightarrow \infty} \frac{n\sqrt{n}}{\sqrt{n^{3}+2}}\\
&=1
\end{align*}
Since $\displaystyle \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ is divergent ($p-$series with $p<1$), then $\displaystyle\sum_{n=1}^{\infty} \frac{n}{\sqrt{n^{3}+2}}{n^{4}-1}$ also divergent, hence $\displaystyle\sum_{n=1}^{\infty}(-1)^{n} \frac{n}{\sqrt{n^{3}+2}}$ is conditionally convergent.