Irrationality of quick convergent series. (English) Zbl 0870.11040

Hančl, Jaroslav

J. Théor. Nombres Bordx. 8, No. 2, 275-282 (1996).

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.

Reviewer: C.Badea (Villeneuve d’Ascq)

MSC:

11J72

Irrationality; linear independence over a field

Keywords:

irrationality; irrational numbers; infinite series

Citations:

Zbl 0770.11036

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