Convergence/Divergence of infinite series $\sum_{n=1}^{\infty} \frac{(\sin n+2)^n}{n3^n}$ $$ \sum_{n=1}^{\infty} \frac{(\sin n+2)^n}{n3^n}$$
Does it converge or diverge? 
Can we have a rigorous proof that is not probabilistic? 
For reference, this question is supposedly a mix of real analysis and calculus.

I propose the following heuristic argument that the series converges:
The natural numbers $n$ are uniformly distributed ${\rm mod}\ 2\pi$. Therefore the expected value of the $n$-th term of the series is
$$a_n:={1\over n}\int_{-\pi}^\pi\left({2+\cos\phi\over 3}\right)^n\ d\phi\ .$$
Now a look at the graphs shows that
$${2+\cos\phi\over 3}\leq e^{-\phi^2/9}\qquad(-\pi\leq\phi\leq\pi)\ .$$
Therefore
$$a_n\leq{1\over n}\int_{-\pi}^\pi e^{-n\phi^2/9}\ d\phi<{1\over n} \int_{-\infty}^\infty e^{-n\phi^2/9}\ d\phi={\sqrt{3\pi}\over n^{3/2}}\ ,$$
which leads to convergence.
