Answer
Diverges
Work Step by Step
Given
$$ \sum_{k=2}^{\infty}\frac{ k^{1/3} }{k^{5/4}-k} $$
Compare with $\displaystyle\sum_{k=1}^{\infty}\frac{1}{k^{11/12}}$, which is a divergent series ( $p-$series with $p<1$) and for $k\geq 2$
\begin{align*}
\frac{ k^{1/3} }{k^{5/4}-k} &\geq \frac{1 }{k^{11/12}}
\end{align*}
Then $\displaystyle\sum_{k=2}^{\infty}\frac{ k^{1/3} }{k^{5/4}-k} $ also diverges.
