Summability characterizations of positive sequences

TL;DR

This paper extends classical summability tests like the Kummer test to better characterize the convergence of positive series, offering new criteria, consequences, and examples.

Contribution

It introduces extensions to the Kummer test and related classical tests, broadening their applicability and providing new insights into series convergence.

Findings

Extended Kummer test for positive series

New criteria for convergence and divergence

Examples illustrating the extended tests

Abstract

In this paper, we propose extensions for the classical Kummer test, which is a very far-reaching criterion that provides sufficient and necessary conditions for convergence and divergence of series of positive terms. Furthermore, we present and discuss some interesting consequences and examples such as extensions of the Olivier's theorem and Raabe, Bertrand and Gauss's test.