Transcendence of some infinite series

TL;DR

This paper uses Roth's theorem to provide a sufficient condition under which an infinite series of positive rational terms is transcendental, also establishing a measure for such series.

Contribution

It introduces a new criterion based on Roth's theorem to determine the transcendence of certain infinite series of positive rational numbers.

Findings

Provides a sufficient condition for transcendence of series of positive rationals

Establishes a transcendental measure for the series

Connects Roth's theorem with transcendence criteria

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 series of positive rational terms is a transcendental number. With the same conditions, we establish a transcendental measure of $ \sum_{n = 1}^{\infty} 1/a_n$.