Sum of a series close to geometric

Question

I'm stuck in finding the sum: \begin{align} S = \sum_{k=2}^n \frac{1}{2^k-1} \end{align} It seems to be close to geometric without being geometric. I wanted to try to transform it to a geometric series but I don't seem to come up with a valid transformation to do it. Anyone has an idea?

Answer 1

$$S = \sum_{k=2}^n \frac{1}{2^k-1}$$ This sum cannot be expressed with a finite number of elementary functions. For a closed form a special function is required, namely the so called q-digamma function $\psi_{q}(x)$ :

http://mathworld.wolfram.com/q-PolygammaFunction.html

$$\sum_{k=2}^n \frac{1}{2^k-1}=\frac{1}{\ln(2)}\left(\psi_{1/2}(n+1)-\psi_{1/2}(2) \right)$$