## 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:

 [1] Badea, C., The Irrationality of Certain Infinite Series, Glasgow. Math J.29 (1987), 221-228. · Zbl 0629.10027 [2] Badea, C., A Theorem on Irrationality of Infinite Series and Applications, Acta Arith. LXII.4 (1993), 313-323. · Zbl 0770.11036 [3] Brun, V., A Theorem about Irrationality, Arch. for Math. og Naturvideskab Kristiana31,, (1910), 3 (German). · Zbl 1234.11092 [4] Erdös, P., Some Problems and Results on the Irrationality of the Sum of Infinite Series, J. Math. Sci.10 (1975), 1-7. · Zbl 0372.10023 [5] Erdös, P. and Strauss, E.G., On the Irrationality of Certain Ahmes Series, J. Indian Math. Soc.27 (1963), 129-133. · Zbl 0131.04902 [6] Fichtengolz, G.M., The Lecture of Differential and Integral Calculations, part I, issue 6, Nauka, Moskva, 1966 (Russian). [7] Hančl, J., Criterion for Irrational Sequences, J. Num. Theory 43 n° 1 (1993), 88-92. · Zbl 0768.11021 [8] Sándor, J., Some Classes of Irrational Numbers, Studia Univ. Babes-Bolyai Math.29 (1984), 3-12. · Zbl 0544.10033
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