Which convergence test to use I need to prove that the following converges: 
$$\sum\limits_{n=0}^{\infty} n^3e^{-n^4}$$
I'm not sure which test to use. I've tried ratio test but it gets messy. I don't think I have any sums to use in the comparison test and the integral test obviously wouldn't work. Thanks

Directly $\;n-$th root test:
$$\sqrt[n]{n^3e^{-n^4}}=\frac{\left(\sqrt[n]n\right)^3}{e^{n^3}}\xrightarrow[n\to\infty]{}0<1$$
and thus the series converges
