Regular Variation and Raabe

TL;DR

This paper unifies classical convergence tests for series, specifically Raabe and Bertrand tests, through the lens of regular variation, revealing their connections and providing a broader theoretical framework.

Contribution

It introduces a unified approach to classical series convergence tests using the theory of regular variation, clarifying their relationships and expanding their applicability.

Findings

Raabe and Bertrand tests are linked to regular variation.

A unified framework for classical convergence tests is established.

The approach enhances understanding of series convergence criteria.

Abstract

There are many tests for determining the convergence or divergence of series. The test of Raabe and the test of Betrand are relatively unknown and do not appear in most classical courses of analysis. Also, the link between these tests and regular variation is seldomly made. In this paper we offer a unified approach to some of the classical tests from a point of view of regular varying sequences.