## On the irrationality of $$\sum (1/(q^ n+r))$$.(English)Zbl 0718.11029

In “Old and new Problems and results in combinatorial number theory” [Enseign. Math. 28 (1980; Zbl 0434.10001)] P. Erdős and R. L. Graham claimed that the irrationality of $$\sum^{\infty}_{n=1}(2^ n-3)^{-1}$$ is unresolved. Using Padé approximation to the q-analogue of log, $L_ q(x)=\sum^{\infty}_{n=1}x/(q^ n-x),\quad | q| >1,\quad x\neq q^ m\quad \quad \quad (m\in {\mathbb{N}})$ it is proved: If q is an integer greater than one and r is a nonzero rational $$(r\neq -q^ m)$$ then $$\sum^{\infty}_{n=1}1/(q^ n+r)$$ is irrational and is not a Liouville number.

### MSC:

 11J72 Irrationality; linear independence over a field