On the irrationality of \(\sum (1/(q^ n+r))\) (Q2639088)

In ``Old and new Problems and results in combinatorial number theory'' [Enseign. Math. 28 (1980; Zbl 0434.10001)] \textit{P. Erdős} and \textit{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.