The Second Raabe's Test and Other Series Tests

TL;DR

This paper surveys classical series convergence tests, introduces an extension of Raabe's Test called the Second Raabe's Test, and proposes further extensions of Gauss' and Kummer's Tests, including proofs and applications.

Contribution

It introduces the Second Raabe's Test and extends classical tests like Gauss' and Kummer's Tests, providing proofs and applications.

Findings

Extension of Raabe's Test to the Second Raabe's Test

Proposed extensions of Gauss' and Kummer's Tests

Provided proofs and a brief application of the Second Raabe's Test

Abstract

The classical D'Alembert's Ratio Test is a powerful test that we learn from calculus to determine convergence for a series of positive terms. Its range of applicability and ease of computation makes this test extremely appealing. However, it admits an inconclusive case when the limiting ratio of the terms equals 1. Several series tests like Raabe's and Gauss' Tests have been proposed in order to address this case. These tests were also generalized by Kummer through Kummer's Test. More recently, a Second Ratio Test was constructed that similarly possessed an inconclusive case. This article will present a survey of existing series tests. Secondly, it will introduce an extension of Raabe's Test to the Second Ratio Test. Thirdly, other extensions of classical tests such as Gauss' Test and Kummer's Test are proposed. Finally, it will also present proofs for the aforementioned tests and a brief application of the Second Raabe's Test.