# Is there a proof for the following series to diverge/converge?

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?

• 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 1 '14 at 23:09
• – dani_s Mar 1 '14 at 23:10
• @dani_s Thanks, this one was linked: jstor.org/discover/10.2307/… 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? – user117449 Mar 1 '14 at 23:16
• @sos440, do you have any evidence for the series to converge for large $b$? How large? – user117449 Mar 1 '14 at 23:20