Answer
Converges conditionally.
Work Step by Step
We have the absolute series
$$\sum_{n=1}^{\infty} |\frac{(-1)^{n-1}}{ n^{2/3}}|=\sum_{n=1}^{\infty} \frac{1}{n^{2/3}}$$
which is a divergent p-series as $p=2/3\lt 1$.
The series $\frac{1}{ n^{2/3}}$ is positive, decreasing and tending toward zero. Thus, the original series converges by the Alternative Series Test.
Therefore, overall the series converges conditionally.
