## A theorem on irrationality of infinite series and applications.(English)Zbl 0770.11036

It is proved a result showing a relationship between the (ir)rationality of a convergent infinite series of positive rationals and some (in)equalities among its terms. The applications include some results on the irrationality of the sum of reciprocals of certain recurrence generated sequences, an improvement of an old result of W. Sierpinski and a generalization of a theorem due to A. Oppenheim concerning the algorithms which are now called Oppenheim expansions.
As a sample, we mention the following consequence of the main result: if the sum of the series $$\sum^ \infty_{n=1}b_ n/a_ n$$, with $$b_ n$$ and $$a_ n$$ positive integers, is a rational number and $$a_{n+1}\geq(b_{n+1}/b_ n)a^ 2_ n-(b_{n+1}/b_ n)a_ n+1$$ for all $$n$$, then we must have equality from a point on.