That response I gave is incorrect. Also it's complicated compared to the following valid solution:
xe^(-x) is convergent
therefore
(n^2) e^(-n^2) is convergent.
Now
(n^2) e^(-n^3)
= (n^2) e^(-n^2) * [e^(n^2 - n^3)]
< (n^2) e^(-n^2) since e^(n^2 - n^3) < 1 when n > 1
Therefore, by the comparison test, the series is convergent.