>Could You give a poof, comment or reference for the inequality as follows:
>
> $$\sum_{k=1}^n(-1)^k\frac{\sin kx}{k^{\alpha}} < 0$$
for all $n=1,2,3,\ldots$ and $0<x<\pi$ and $\alpha \ge 1$

* See also:

https://mathoverflow.net/questions/303759/sum-k-1n-frac-sin-kxk-alpha-0-quad-textfor-all-n-1-2-3-ldots-te?noredirect=1&lq=1