Closed form of $\sum_{n=1}^{\infty} \frac{1}{2^n(1+n^2)}$ How would you recommend me to tackle the series 
$$\sum_{n=1}^{\infty} \frac{1}{2^n(1+n^2)}$$?
Can we possibly express it in terms of known constants? What do you think about it?

Mathematica yields
Sum[1/(2^n (1 + n^2)), {n, \[Infinity]}]

giving $$\left(\frac{1}{4}-\frac{i}{4}\right) (\, _2F_1(1,1;2-i;-1)+i \, _2F_1(1,1;2+i;-1))\approx0.318057$$
where $_2F_1$ is the confluent 2F1 hypergeometric function, which agrees with numerical summation.
As for how one could obtain this result by hand, perhaps another user could supply that answer?
