Can the sum $\sum_{k=1}^\infty (1/k)^{3/2}\sin(kx)$ be evaluated using Fourier series or otherwise?

Question

I have to compute this sum, and I was wondering if it can be evaluated using Fourier series. It seems familiar to me but have forgotten the Fourier tricks I used in the past, so time for revision.

$$\sum_{k=1}^\infty (1/k)^{3/2}\sin(kx)$$

Answer 1

Your series, as it stands, is already a Fourier series.

One may express it in terms of the polylogarithm function $\text{Li}_s(\cdot)$, for $x \in \mathbb{R}$, one has

$$ \sum_{k=1}^\infty \frac1{k^{3/2}}\sin(kx)=\Im\sum_{k=1}^\infty \frac{e^{ikx}}{k^{3/2}}=\Im\: \text{Li}_{3/2}(e^{ix}). $$

You will get different representations of $\text{Li}_s(\cdot)$ here.