# Show that $\sum_{n=1}^{\infty}\sin \left(\frac{n\pi}{3} \right)\frac{1}{n^r}$ diverges for $0<r<1$

I would like to show that $$\displaystyle \sum_{n=1}^{\infty}\sin \left(\frac{n\pi}{3} \right)\frac{1}{n^r}$$ diverges when $$0. I'm having a hard time doing this though. It seems that p-series would obviously be related, but I can't make any comparison work.

Additionally, how could I show that the sum converges for $$r=1$$?

• You might want to look up Dirichlet's test for convergence. – zhw. Oct 8 '18 at 20:08
• @wesley The series converges for $r>0$. – Mark Viola Oct 8 '18 at 20:18
• @MarkViola By Dirichlet's test? – Wesley Oct 8 '18 at 22:01
• Yes. Or use summation by parts. – Mark Viola Oct 8 '18 at 22:12