Irrationality of quick convergent series. (English) Zbl 0870.11040

Using elementary methods, the reviewer proved in 1987 the following criterion for the irrationality of infinite series : if \((a_n)\) and \((b_n)\) are two sequences of positive integers such that \(b_{n+1} > (b_n^2 - b_n)a_{n+1} + 1\), then the sum of the series \(\sum_{n=1}^{\infty} b_n/a_n\) is irrational.
This result was generalized and improved by several authors since 1987 ; cf. C. Badea [Acta Arith. 63, 313 - 323 (1993; Zbl 0770.11036)] and the references cited therein. A new generalization is given in the paper under review by replacing the above condition by a sequence of conditions depending on a parameter \(m\). For \(m = 0\) we obtain the 1987 result.


11J72 Irrationality; linear independence over a field


Zbl 0770.11036
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