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?