How to prove the convergence of the series $\sum\frac{(3n+2)^n}{3n^{2n}}$ I need to prove the convergence of this series:
$$\sum\limits_{n=1}^\infty\frac{(3n+2)^n}{3n^{2n}}$$
I've tried using Cauchy's criterion and ended up with a limit of $1$, but I already know it does converge.
How am I suppose to prove it? Note that I've also tried the integral test and was not able to solve.

The root test applies nicely here. Taking $\displaystyle a_n=\left(\frac{3\,n+2}{n^2}\right)^n$ gives 
$$
\lim_{n\to\infty}\lvert a_n\rvert^{1/n}
= \lim_{n\to\infty}\frac{3\,n+2}{n^2} =0<1
$$
This proves that $\sum a_n$ converges so your series $\frac{1}{3}\sum a_n$ converges.
