TITLE: infinite series of the form $\sum\limits_{k=1}^{\infty}\frac{1}{a^{k}+1}$
QUESTION [9 upvotes]: Is there a method for evaluating infinite series of the form:
$\displaystyle\sum_{k=1}^{\infty}\frac{1}{a^{k}+1}, \;\ a\in \mathbb{N}$?.  For instance, say $a=2$ and we have
$\displaystyle\sum_{k=1}^{\infty}\frac{1}{2^{k}+1}$.
I know this converges to $\approx .7645$, but is it possible to find its sum using some clever method?.
It would seem the Psi function is involved. I used the sum on Wolfram and it gave me
$$-1+\frac{{\psi}_{1/2}^{(0)}\left(1-\frac{\pi\cdot i}{\ln(2)}\right)}{\ln(2)}$$
I am familiar with the Psi function, but I am unfamiliar with that notation for Psi. What does the 1/2 subscript represent?

REPLY [4 votes]: Not entirely relevant, but it's known that these numbers are irrational.
Peter B. Borwein, On the irrationality of $\sum_{n\gt0}(1/(q^n+r))$, 
J. Number Theory 37 (1991), no. 3, 253–259, MR1096442 (92b:11046) proves the sum is irrational when $r$ is rational and $q\ge2$ is an integer. Could be worth a look. 
EDIT: As Dan notes, $r$ can't be zero. Also, it can't be of the form $-q^m$, lest we find ourselves dividing by zero.