Decide if the series $ \sum_{n=1}^{\infty} \sin\left( \frac{n\pi}{6}\right)$ converges or not.

I've tried to use the ratio test but I had no success. I don't see how other convergence tests could work.

Thanks for your help!

  • 3
    $\begingroup$ Can you think of any two subsequences of $\{\sin(n\frac {\pi}6)\}_n$ that converge to disparate values? $\endgroup$ – abiessu Oct 17 '13 at 15:10
  • $\begingroup$ @abiessu yes, but I don't get why is this important since I am talking about the serie $\endgroup$ – Giiovanna Oct 17 '13 at 15:31

Hint: What is $\lim_{n\to\infty}\sin\left(\frac{n\pi}{6}\right)$?

  • $\begingroup$ It does not exists, but I don't see how this can help $\endgroup$ – Giiovanna Oct 17 '13 at 15:30
  • 3
    $\begingroup$ @user2768645: If it doesn't exist, then it must not equal $0$... $\endgroup$ – Clayton Oct 17 '13 at 15:31
  • $\begingroup$ Ok, is this a criterion of convergence? $\endgroup$ – Giiovanna Oct 17 '13 at 15:35
  • 2
    $\begingroup$ It is the first test for convergence taught in most calculus courses. $\endgroup$ – Clayton Oct 17 '13 at 15:38
  • $\begingroup$ I didn't know it. Thanks for your attention! $\endgroup$ – Giiovanna Oct 17 '13 at 15:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.