## 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.

### MSC:

 11J72 Irrationality; linear independence over a field

### Keywords:

irrationality; irrational numbers; infinite series

Zbl 0770.11036
Full Text:

### References:

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