Some classes of irrational numbers. (English) Zbl 0544.10033

Sándor, József

Stud. Univ. Babeş-Bolyai, Math. 29, 3-12 (1984).

Let \((n_ k)\) be a sequence of positive integers satisfying \[ (1)\quad 1<n_ 1<n_ 2<\ldots,\quad (2)\quad \overline{\lim}_{k\to \infty}(n_ k/n_ 1...n_{k-1})=\infty,\quad (3)\quad \underline{\lim}_{k\to \infty}(n_ k/n_{k-1})>1. \] Then P. Erdős showed that \(\sum^{\infty}_{k=1}1/n_ k\) is irrational. The author extends this to cover the irrationality of \(\sum^{\infty}_{k=1}m_ k/n_ k\) where \(m_ 1,m_ 2,\ldots\) is a suitable sequence of positive integers. He then extends a result of G. Cantor on the irrationality of infinite products. Finally he extends an irrationality result of T. Estermann. In connection with the last result one may look into M. Ram Murty and V. Kumar Murty [Can. Math. Bull. 20, 117–120 (1977; Zbl 0366.10028)].

Reviewer: K. Ramachandra

MSC:

11J72

Irrationality; linear independence over a field

Keywords:

irrational numbers; sum of quotients; sequence of positive integers; infinite products