# Can this series be expressed in closed form, and if so, what is it?

Can this series be expressed in closed form, and if so, what is it? $$\sum_{n=1}^\infty\frac{1}{9^{n+1}-1}$$

• I think there is a closed form for such things using the digamma function, if that qualifies as closed form for you. Sep 26, 2012 at 23:27
• @AndréNicolas well, I suppose it's better than nothing. How do you express it in terms of the digamma function? Sep 27, 2012 at 0:49
• Erdos asked whether numbers of that sort are irrational. Peter Borwein settled that question in a paper in the Journal of Number Theory, volume 37 (1991), no. 3, pages 253–259. Sep 27, 2012 at 0:51
• @Navin: It was sort of a guess. Go to Maple, or Mathematica, or Alpha and it will tell you. Sep 27, 2012 at 1:22
– mick
Dec 13, 2013 at 23:33

Let us denote by $f$ the following function
$$f(a):=\sum_{n=1}^\infty\frac{1}{a^{n+1}-1}.$$
$$f(a) = \sum_{n=1}^\infty\frac{1}{a^{n+1}-1}= \frac{1}{1-a}-\sum_{n=1}^\infty\frac{1}{1-a^n}\stackrel{\left(\spadesuit\right)}{=} \frac{1}{1-a}-\frac{\psi_{1/a}(1)+\ln(a-1)+\ln(1/a)}{\ln(a)},$$
where $\psi_q(z)$ denotes the $q$-polygamma function, and in $\left(\spadesuit\right)$ we used the equation $(4)$ from here.
$$f(9) = \frac78 - \frac{\ln(8)}{\ln(9)} - \frac{\psi_{1/9}(1)}{\ln(9)} \approx 0.014045117662188129358728474369089\dots$$ Note that $f(2)=\mathcal{C}_{\textrm{EB}}-1 \approx 0.60669515241529\dots,$ where $\mathcal{C}_{\textrm{EB}}$ is the Erdős–Borwein constant.