Irrational Sequences of Cauchy Type
by
Jaroslav Hancl
University of Ostrava

Erdös in his paper [1] introduced the so-called irrational sequences in the following way.

Definition: Let {a_n} be a sequence of positive numbers. If there is a sequence {c_n} of positive integers such that

\infty å n=1  1/(ancn)

is a rational (algebraic) number then {a_n} is an irrational (a transcendental) sequence. Otherwise {a_n} is a rational (algebraic) sequence.

This contribution deals with the following criterion for irrational sequences of Cauchy type.

Theorem: Let {a_n} and {b_n} be two sequences of positive integers such that a_n >= 2²ⁿ and b_n <= 2^2n-\surd{3n}. Then the sequence

{( n Õ i=1  ai)/bn }

is irrational.

A consequence of this theorem is a famous Erdös result that the sequence {2²ⁿ} is irrational.

[1]   Erdös P.: Some Problems and Results on the Irrationality of the Sum of Infinite Series, J. Math. Sci. 10 (1975), 1-7.

(AMS Class.: 11J72)

Date received: April 19, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacf-35.