Is the series $S_n=\sum_{n=1}^{\infty}\frac{(-1)^n \sin n x }{n^{\log_e n}}$ converges for all $x \in \mathbb{R}?$
Since $\log_e(n) > 2$ for $n > e^2$ and $|(-1)^n \sin(nx)| \le 1$, $\left|\frac{(-1)^n \sin n x }{n^{\log_e n}}\right| \lt \frac1{n^2} $ so the sum converges.