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

First note that
$$\sum\limits_{n=0}^{\infty} \left|\frac{n^3}{e^{n^4}}\right|=\sum\limits_{n=1}^{\infty} \left|\frac{n^3}{e^{n^4}}\right|\leq\sum\limits_{n=1}^{\infty} \left|\frac{n^3}{n^5}\right|=\sum\limits_{n=1}^{\infty} \left|\frac{1}{n^2}\right|$$
Therefore
$$\sum\limits_{n=0}^{\infty} \frac{n^3}{e^{n^4}}=\mbox{absolutely convergent}$$
