I would like to show that $\displaystyle \sum_{n=1}^{\infty}\sin \left(\frac{n\pi}{3} \right)\frac{1}{n^r}$ diverges when $0<r<1$. 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 $r=1$?

  • $\begingroup$ You might want to look up Dirichlet's test for convergence. $\endgroup$ – zhw. Oct 8 '18 at 20:08
  • 2
    $\begingroup$ @wesley The series converges for $r>0$. $\endgroup$ – Mark Viola Oct 8 '18 at 20:18
  • $\begingroup$ @MarkViola By Dirichlet's test? $\endgroup$ – Wesley Oct 8 '18 at 22:01
  • $\begingroup$ Yes. Or use summation by parts. $\endgroup$ – Mark Viola Oct 8 '18 at 22:12

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