Test the convergence of the series $\sum_{n=1}^\infty \frac{n^n}{3^n n!}$
I know that if the nth term tends to $\infty$ then the series is divergent and if it is tends to 0 it is convergent . Also I'm familiar with some test e.g. Ratio test, d'Alembert test, comparison test etc. But I could not solve it in proper way. I know as $n$ increases $n^n$ increases more rapidly than $n!$ or $3^n$ but no idea when they both are multiplied