TITLE: Convergence of a series with repeated sines
QUESTION [34 upvotes]: Show that the series 
$$
\sum_{n=1}^\infty \frac{\sin\big(\sin(n)\big)}{n},
$$
converges.
More generally, show that for every $k\in\mathbb N$ the series
$$
\sum_{n=1}^\infty \frac{\sigma_k(n)}{n},
$$
converges, where $\sigma_1(x)=\sin(x)$ and $\sigma_{k+1}(x)=\sin\big(\sigma_k(x)\big)$.
Note. I am looking for an elementary proof, if such is available. I do know that such elementary proof exists, for $k=2$, since that case was  qualifying exam (University of Adelaide) a few years ago. On the other hand, I know of a non elementary proof (for $k$ general), using the nontrivial fact that $\dfrac{1}{2\pi}$ has finite irrationality measure. I imagine that any proof would have as a first step establishing the fact that the sequence
$$
\sum_{j=1}^n \sigma_k(j), \quad n\in\mathbb N, 
$$
is bounded.
Update. Actually, using advanced tools (i.e., irrationality measure), 
it turns out that even
$$
\sum_{n=1}^\infty \frac{\sigma_k(n)}{n^a},
$$
converges, for every $a>0$, and $k\in\mathbb N$,
while the same tools can not determine whether
$$
\sum_{n=1}^\infty \frac{\sin\big(\sin(\beta n)\big)}{n^a},
$$
converges, for every $\beta$.

REPLY [2 votes]: If $f(x)$ is an odd continuous function with range $[-a,a]$, and $f(x) \leq |x|$,  then $f^k(x)$ has the same property for all $k \geq 1$.  
If a sequence $a_n$ has a symmetric (even) frequency distribution measure in the interval $[-a,a]$, then $f^k(a_n)$ has the same property.
Based on this, and taking $a_n$ to be the value of $(n \mod 2\pi)$ in $(-\pi,\pi)$, by Abel summation the $\sum \frac{f^k(a_n)}{n}$ converges if $s_n = f^k(a_1) + \dots + f^k(a_n)$ is $O(n^c)$ for $c < 1$.  Uniform distribution modulo $1$ gives only $s_n = o(n)$ which is an epsilon less than what we need.  
The difference $d_n = |\frac{s_n}{n} - \int_{-\pi}^{\pi} f^k(x) dx|$ converges to $0$ by uniform distribution (applied to $f^k \circ g$ for $g$ that re-coordinatizates the interval to make the distribution uniform). For $f^k$ of bounded variation,  we are asking a special case of the discrepancy problem for sequence $a_n$ : is there a bound $d_n = O(n^{-u})$ for $u > 0$?  
This power-of-$n$ improvement in the convergence is true for any sequence with a positive irrationality measure.  Maybe a sledgehammer but it shows the problem does not need any special property of $f(x)=\sin x$ except that $f^k$ have bounded variation (which holds if $f'$ exists and is continuous).