Does this series converge? Root test and ratio test are inconclusive.
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1Related question : http://mathoverflow.net/questions/24579/convergence-of-sumn3-sin2n-1 – Daniel Soltész Jun 24 '14 at 14:14
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1@hallaplay835, yes it exists, check this link to be sure of it : http://math.stackexchange.com/a/736374/140057 – Fabien Jun 24 '14 at 14:16
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2This is an exercise in an introduction to series text. There's no way this was supposed to be solved using irrationality measure of $\pi$. – dazedviper Jun 24 '14 at 15:15
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1@hallaplay835 can you list the theorems you know about series ? – Gabriel Romon Jun 24 '14 at 17:10
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Using the same arguments as in the two answers of Does $\sum_{n=1}^\infty \frac{1}{n! \sin(n)}$ diverge or converge? we can compare the series $\sum^\infty_{n=1}\left|\frac{1}{3^n\sin n}\right|$ to $\sum^\infty_{n=1}\frac{n^7}{3^n}$, which certainly converges by the ratio test, so that the original series converges (absolutely).
