infinite series of the form $\sum\limits_{k=1}^{\infty}\frac{1}{a^{k}+1}$

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?

Solution 1:

What you want to look at is Lambert Series. Notice that in the expression Wolfram Alpha spat out, the $\frac{-\pi i}{\log 2}$ is simply there to turn the positive sign into a negative one so that we are dealing with Lambert Series.

Theses series are power series where the coefficients are given by Dirichlet convolution, so they are often related to multiplicative functions.

Let $$L(q)=\sum_{n=1}^\infty a_n \frac{q^n}{1-q^n}=\sum_{n=1}^\infty \left(\sum_{d|n} a_d \right) q^n.$$

Then we can rewrite a series related to yours above: $$\sum_{n=1}^\infty \frac{1}{2^n-1}=\sum_{n=1}^\infty \frac{d(n)}{2^n}.$$

However, if the coefficients $a_n$ are functions which become nice when convolved with $1$ we can get something different. For example $$\sum_{n=1}^\infty \mu(n)\frac{q^n}{1-q^n}=q.$$

Solution 2:

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.