Related to an open problem about another series.

Set

$$A= \sum_{n=1}^\infty \frac{\psi^{(1)}(1-n/\pi)}{n^3}$$

where $\psi^{(n)}(k)$ is the polygamma function.

Does $A$ converge?

The related series $\sum_{n=1}^\infty \frac{\psi^{(1)}(n/\pi)}{n^3}$ converges.

According to Maple 13: $\lim_{n \to \infty} \frac{\frac{\psi^{(1)}(1-n/\pi)}{n^3}}{\frac{1}{n^2}}=0$. If this is true it appears to give convergence. The series expansion at infinity involves $\cot$ so I suppose Maple's result might be wrong.

What other CASes say about the series?

I suppose $A$ converges iff $\sum_{n=1}^\infty \cot^2(n)/n^3$ converges.

Experimentally $A$ is about $278.73$.