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.

Let's use the $n$th root test.
$$
\left|\frac{(3n+2)^n}{3n^{2n}}\right|^{1/n} = \frac{3n+2}{3^{1/n}n^2} \to 0.
$$
Since $0<1$, the series converges absolutely.
