I wanted to see if the function SumConverge worked for the following summation: $$\sum_{n=1}^{\infty}\frac{\sin^2(n)}{n^p}$$ We can clearly see that for $p>1$ the series converges and for $p\le1$ diverges by the comparison test. I tried:

SumConvergence[(Sin[n])^2/n^4, n] 

However, I do not get an evaluation. Why so? $$$$I also tried $$\sum_{n=1}^{\infty}\left|\frac{\sin(n)}{n^p}\right|$$ Whose convergence or divergence can by found by Dirichlet's Test. However mathematica seems to not understand such input. I tried:

SumConvergence[Abs[(Sin[n])/n], n]