Answer
converges conditionally
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt[3]{n}+2 n}$$
Since
$$\sum_{n=1}^{\infty}\left|a_{n}\right|=\sum_{n=1}^{\infty} \frac{1}{n^{1 / 3}+2 n}$$
Using the comparison test, we compare with $ \sum \frac{1}{n} $, a divergent series:
\begin{align*}
\lim\limits_{n \to \infty}\frac{n}{n^{1 / 3}+2 n}=1
\end{align*}
Hence $\sum_{n=1}^{\infty}\left|a_{n}\right| $ diverges.
On the other hand, since $ a_n =\dfrac{1}{\sqrt[3]{n}+2 n} $ is decreasing and $\lim_{n\to \infty } a_n=0$, then $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt[3]{n}+2 n}$ converges. Hence, $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{1 / 3}+2 n}$ converges conditionally.
