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?

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 $b$.