Examine the convergence of the sequence $a_n=\frac{(n^2+356)\arctan{n}}{(n!)^2}$. Do sequences $na_n$ and $(n!)a_n$ converge?

Question

I thought I should give it a try with Cauchy's criterion, or try to find the limit using Stolz-Cesáro theorem (we're not supposed to use l'Hopital's rule yet), but that doesn't seem to go anywhere.

Answer 1

Note that $|\tan^{-1}n|\leq\pi/2$ and so $(n!)|a_{n}|\leq\dfrac{\pi}{2}\cdot\dfrac{n^{2}+356}{n!}\rightarrow 0$. And we have $n|a_{n}|\leq n!|a_{n}|$.