Consider the following infinite series:
$\text{f} \left( x \right) =\displaystyle \sum \limits_{n=1}^{\infty} \frac {\sin \left( n x\right)}{n^2}$
We know that $\text{f} \left( x \right)$ is convergent by the squeeze theorem. The graph of $\text{f} \left( x \right)$ is shown below:
Is there a way to express this periodic function in a closed-form expression, rather than a sum?
Specifically, I am wondering if there is a way to represent $\text{f} \left( x \right)$ in the form of something like $k\cdot \text{g} \left( \sin \left( x \right) \right)$, or $k \cdot \sin \left( \text{g} \left( x \right) \right)$.
This is the famous Clausen function.