Answer
diverges
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{1}{\sqrt[3]{n}(1+\sqrt{n})} $$
We use the limit comparison test with $ \sum \dfrac{1}{n^{5/6}}$, a divergent series:
\begin{align*}
\lim_{n\to \infty } \frac{a_n}{b_n} &=\lim_{n\to \infty } \frac{n^{5/6}}{\sqrt[3]{n}(1+\sqrt{n})} \\
&= \lim_{n\to \infty } \frac{1}{ 1/n^{1/2} +1 }\\
&=1
\end{align*}
Then the given series is also diverges.
