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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!

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

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  • $\begingroup$ It does not exists, but I don't see how this can help $\endgroup$ – Giiovanna Oct 17 '13 at 15:30
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    $\begingroup$ @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
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    $\begingroup$ 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

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