On the transcendence of some operations of infinite series

TL;DR

This paper uses Roth's theorem to establish conditions under which sums, products, and quotients of certain Liouville series are transcendental, and provides an approximation measure for these numbers.

Contribution

It introduces a sufficient condition for the transcendence of operations on Liouville series based on Roth's theorem, advancing understanding of their algebraic properties.

Findings

Sufficient condition for transcendence of series operations

Application of Roth's theorem to Liouville numbers

Establishment of an approximation measure for these numbers

Abstract

In the present paper and as an application of Roth's theorem concerning the rational approximation of algebraic numbers, we give a sufficient condition that will assure us that a sum, product and quotient of some series of positive rational terms are transcendental numbers. We recall that all the infinite series that we are going to treat are Liouville numbers. At the end this article, we establish an approximation measure of these numbers.