Hančl, Jaroslav Irrationality of quick convergent series. (English) Zbl 0870.11040 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) Cited in 2 Reviews MSC: 11J72 Irrationality; linear independence over a field Keywords:irrationality; irrational numbers; infinite series Citations:Zbl 0770.11036 PDF BibTeX XML Cite \textit{J. Hančl}, J. Théor. Nombres Bordx. 8, No. 2, 275--282 (1996; Zbl 0870.11040) Full Text: DOI Numdam EuDML EMIS OpenURL 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.