Answer
Conditionally convergent
Work Step by Step
Let $b_n=\frac{1}{\sqrt[3]{n^2}}$.
We have:
$\sum_{n=1}^{\infty}\left|\frac{(-1)^{n-1}}{n^{2/3}}\right|=\sum_{n=1}^{\infty}\frac{1}{n^{2/3}}$
This is a $p$-series with $p=2/3<1$, so the series diverges, which means the given series in not absolutely convergent.
We have:
$b_n>0$
$\lim_{n\rightarrow\infty}b_n=0$
$b_{n+1}=1/(n+1)^{2/3}<1/n^{2/3}=b_n$
We apply the Alternating Series Test and we obtain that the given series is conditionally convergent.
