Find the range of values $p$ such that the series $$\sum_{n=1}^\infty (-1)^n \sin(\log(n))\cdot \frac{1}{n^p}$$ converges.
Context: this question was on an undergraduate real analysis exam at a university in China.
My intuition is that it should converge for $p>1$ and diverge for $p<0,$ just based on the alternating series test, but I am not sure about the values in the middle.
How would one go about establishing the convergence of such a sum?