Prove that:$$\sum_{k=1}^{\infty}\frac{\sin\left(kx\right)}{k^2}$$is not able to be expressed using elementary functions.
When plugging this into WolframAlpha it returns: $$\frac 12i(\mathrm{Li}_2(e^{-ix})-\mathrm{Li}_2(e^{ix}))$$ Where $\mathrm{Li}_n(z)$ is the polylogarithm.
I've never hear of the polylograithm before, but after reading about it, I have a vague idea how WolframAlpha got there, however I don't know how to show that $\mathrm{Li}_n(z)$ is non-elementary. My other idea would be to turn the sum into an integral using: $$\int_0^xf(t)dt = \lim_{n\to\infty}\sum_{i=1}^nf\left(i\frac xn\right)\frac xn$$