I want to prove a series converges absolutely I want to show that:
$\sum \limits_{n=1}^\infty \cfrac{n^2 + \cos(n)}{e^{n^3}}$ converges absolutely.
Now, here is what I have done: 
$\sum \limits_{n=1}^\infty \cfrac{n^2 + \cos(n)}{e^{n^3}} \leq \cfrac{n^2 + n^2}{e^{n^3}} = \cfrac{2n^2}{e^{n^3}}$
Now here I was told by a friend to use the integral test, but I have no idea how to apply this. Advice please, and then I will post up my attempt. Thank you.

The ratio test is much simpler to use here. You have $$n \leq n^3 \implies e^{-n^3} \leq e^{-n}.$$ Now apply the ratio test to $\dfrac{2n^2}{e^n}$ to get the answer (the ratio converges to $e^{-1} < 1$).
