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.


11J72 Irrationality; linear independence over a field


Zbl 0434.10001
Full Text: DOI


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