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 knew this looked familiar. User Unoqualunque located the key reference. Here is a more recent reference that provides a fairly general approach:

Enrico Laeng, Vittorino Pata. A convergence–divergence test for series of nonnegative 
  terms, Expositiones Mathematicae 29 (4), (2011) 420–424. MR2861768 (2012m:40002).

The authors discuss a test that does not require monotonicity of the sequence, and instead focuses on how "clustered" we find similar terms within the sequence. 
They highlight that their test applies to show that
 $$ \sum_{n=1}^\infty\frac1{n^{2+\cos n}} $$
diverges, while 
 $$ \sum_{n=1}^\infty\frac1n\left(\frac{2+\cos n}3\right)^n $$
converges. They say:

Case (i) has been recently addressed, in (Revue de la filière Mathématique (RMS) 119 (2008–2009), 3–8), where the authors give a proof that was (in their own words) at the frontier between analysis and number theory. Case (ii) apparently originated in a curious way: it was proposed in a calculus exam by mistake, and remained open for a long time thereafter. A solution was devised only ten years later (SIAM Problems and Solutions (2009)), once again by means of quite sophisticated tools. 

As is to be expected, the test is very general but a bit cumbersome to state:
Let $(c_n)_{n\ge1}$ be a sequence of nonnegative terms such that $\sum_n c_n<+\infty$. Let $(a_n)_{n\ge1}$ be a series of nonnegative terms. Then:


*

*The series $\sum_n a_n$ converges if $(na_n)_{n\ge1}$ is a bounded sequence, and there exist $\rho,\vartheta\ge0$ and $\varepsilon\in(0,1]$ such that 
$$ |\{p \in\mathbb N\mid 1\le p\le m\mbox{ and }a_{n+p} > c_n \}| \le \rho m^{ 1 −\varepsilon} $$ 
for every $m$ sufficiently large, and every $n\ge m^\vartheta$.

*The series $\sum_n a_n$ diverges if there exist $\omega> 0$ and $\lambda\ge0$ such that the inequality 
$$ \max_{1\le p\le m} a_{km+p}\ge \frac{\omega}{(km+m)^{1+\lambda/m}} $$ 
holds for infinitely many $m$ and every $k$. 
To apply the test to the series above, one needs to know something about rational approximations to $\pi$ (naturally). Actually, the authors show that to apply the test to show the divergence of the first series only requires that $\pi$ is irrational, and to show the convergence of the second series only needs that $\pi$ is not a Liouville number. The paper is reasonably self-contained. 
