The Case for Raabe's Test

TL;DR

This paper advocates for greater recognition of Raabe's Test in calculus and analysis, highlighting its connections to p-series and its utility in simplifying convergence assessments.

Contribution

It introduces new results on Raabe's Test, emphasizing its pedagogical value and its relation to other comparison tests like the Ratio and Root Tests.

Findings

Raabe's Test can be viewed as an implicit p-series comparison.

It simplifies the determination of conditional convergence.

The paper advocates for including Raabe's Test in undergraduate curricula.

Abstract

Among the techniques for determining the convergence of a series, Raabe's Test remains relatively unfamiliar to most mathematicians. We present several results relating to Raabe's Test that do not seem to be widely known, making the case that Raabe's Test should be featured more prominently in undergraduate calculus and analysis courses. In particular, we demonstrate that Raabe's Test may be viewed as an implicit comparison with a $p$-series, in the same manner that the Ratio Test and the Root Test constitute an implicit comparison with a geometric series. Moreover, Raabe's Test can sometimes simplify the process for determining conditional convergence.