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I was wondering weather the series $$\sum_{n=1}^{\infty} \frac {\tan(n)} {n^b}$$ diverges or converges whenever $b \geq 1$ is an integer. Does anyone have a proof for either statement? Does it converge for some positive integers but not for others? In that case, which are they?

Michael Hardy
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    It is closely related to the irrationality measure $\mu$ of $\pi$, as $\mu$ determines the speed at which $\tan n$ grows. Indeed, it is well-known that $\mu < \infty$, so that the series converges for large $b$. – Sangchul Lee Mar 01 '14 at 23:09
  • See http://math.stackexchange.com/questions/470527/does-the-sum-sum-n-1-infty-frac-tan-nn2-converge – dani_s Mar 01 '14 at 23:10
  • @dani_s Thanks, this one was linked: http://www.jstor.org/discover/10.2307/2690793?uid=3737512&uid=2129&uid=2&uid=70&uid=4&sid=21100858189611

    I dont feel like signing up in order to read the paper, do you know if the paper is relevant? In that case, can it be found somewhere else?

     –  Mar 01 '14 at 23:16
  • @sos440, do you have any evidence for the series to converge for large $b$? How large? –  Mar 01 '14 at 23:20
  • @GeorgeMouselli No I don't know, the answer to the question seems to be far from trivial though – dani_s Mar 01 '14 at 23:27
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    Any $b > \mu(\pi^{-1}$$ works. – Sangchul Lee Mar 01 '14 at 23:31
  • I made a mistake in my previous post. – Mustafa Said Mar 01 '14 at 23:34
  • get a look here : http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.112.5431&rep=rep1&type=pdf –  Mar 04 '14 at 23:07

1 Answers1

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The series certainly diverges for $b=1$ and converges for $b\ge 8$, as shown in this article by Sam Coskey. Moreover, it "stays very small for a very long time" when $b=2$.

In one of the answers to this question, it is argued that the series will converge if $b > \mu(\pi^{-1})$, where $\mu$ is the irrationality measure. The irrationality measure of $1/\pi$ is the same as the irrationality measure of $\pi$, which has recently received an improved upper bound of $7.6063$ (Salikhov 2008; see MathWorld's article for full reference). While this argument reproduces the already-known result (convergence for $b\ge 8$), it shows that a tighter upper bound on $\mu(\pi)$ would immediately imply convergence for smaller values of $b$.

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