Erdős [Er48] proved that $\sum \frac{1}{2^n-1}=\sum \frac{\tau(n)}{2^n}$ is irrational (where $\tau(n)$ is the divisor function). He notes [Er88c] that $\sum \frac{1}{2^n+t}$ should be transcendental for every integer $t$ (presumably intended with the obvious exception $t=0$).

This was proved by Borwein [Bo91], who more generally proved that, for any integer $q\geq 2$ and rational $r\neq 0$ (distinct from $-q^n$ for all $n\geq 1$), the series\[\sum_{n=1}^\infty \frac{1}{q^n+r}\]is irrational.

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This page was last edited 29 September 2025.

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