Criterion for irrational sequences. (English) Zbl 0768.11021

Hančl, Jaroslav

J. Number Theory 43, No. 1, 88-92 (1993).

The main result of the paper is the following. Let \((a_ n)\) and \((b_ n)\) be two sequences of positive integers such that \(a_ n \geq r^{2^ n}_ n\) and \(b_{n+1} \leq r^ B_ n\) hold for all sufficiently large \(n\). Here \((r_ n)\) is a nondecreasing sequence of positive reals tending to infinity and \(B\) is a positive integer. Then the sum of the series \(\sum_{n \geq 1}b_ n/a_ n\) is an irrational number.

Reviewer: C.Badea (Orsay)

MSC:

11J72

Irrationality; linear independence over a field

Keywords:

irrational numbers; series of rational numbers; irratinal sequences