# Determine the convergence/divergence of the following series

I tried using DCT and LCT, but in both cases the comparison test fails. Below is the series: $$\sum_{n=1}^{\infty} (-1)^n \frac {\sin(n)}{n^{0.9}}$$.

Using the absolute convergence test is the first step but from thereon proving that the series is convergent/divergent is tricky. The implementation of the maclaurin series didn't work out either. I would love to know about any other method as well.

Hint: This series is the imaginary part of $\sum\limits_{n\geqslant1}n^{-a}z^n$ for some real positive $a$ and some complex $z\ne1$ such that $|z|=1$. Does this ring a bell?