You should try to apply the following theorem:

Alternating series test: A series of the form ∑[ (−1)^n] an (with an ≥ 0) is called alternating. Such a series converges if the sequence an is monotone decreasing and converges to 0. The converse is in general not true.

You noticed that bn= 1/n^(2/3) converges to 0.

All you have to do is to show that is also decreasing.

Use a ratio test:bn+1/bn=n^2/3/(n+1)^2/3=(n/n+1)^2/3<1, so decreasing.