TITLE: Please help me to find the sum $\sum\limits_{n\geq1} \frac{\sin(nx)}{n^2}$
QUESTION [9 upvotes]: I must show that $\sum\limits_{n\geq1} \frac{\sin(nx)}{n^2}$ converges $\forall x \in {R}$. Then  if $f(x)=\sum\limits_{n=1}^\infty f_n(x)$, I must prove that  $f(x)$ is continuous for $x\in [0, \pi]$  and  $\int\limits_0^\pi f(x)=2\sum\limits_{n=1}^\infty \frac{1}{(2n-1)^3}$.
I used Weistrass M-test: $\sum\limits_{n\geq1} \frac{\sin(nx)}{n^2}$ converges uniformly because $|\frac{\sin(nx)}{n^2}|\le \frac{1}{n^2}, \forall x\in R$ and $\sum \frac {1}{n^2}$ converges.
I found somewhere that $f(x)=\lim\limits_{n\to\infty}f_n(x)$. Where does it come from? I don't understand. How $f(x)$ looks like? Without $f(x)$ I don't know how to prove the last equality from exercise:( 
Later edit: If $ \sum\limits_{n\geq1}f_n\to f$ uniformly , then $f_n\to f$ (punctual converges to $f $- I hope this is correct translation) which means $f_n(x)\to f(x)$ when $n\to \infty ?$ 
I hope someone could help me again...

REPLY [4 votes]: Differentiate twice, get the series for $-\dfrac{1}{2 \tan\frac{x}{2}}$.  So now integrate this twice... Of course that is not elementary, but still...