A connection between tests for absolute convergence of infinite series, or how to be fair

TL;DR

This paper explores the relationships among classical convergence tests for infinite series, introduces intermediate Power Mean Tests, and advocates for their inclusion in calculus education to promote fairness and comprehensive understanding.

Contribution

It introduces Power Mean Tests as intermediate convergence tests, connects them with classical tests, and argues for their pedagogical inclusion in calculus courses.

Findings

Power Mean Tests are stronger than Ratio Test but weaker than Root Test.

All these tests relate to the Generalized f-mean Test.

An example series favors Arithmetic Mean Test for convergence verification.

Abstract

The Ratio Test and the Root Test for absolute convergence/divergence of series of numbers $\sum_{n=0}^{\infty}a_n$ are frequently discussed and proved independently in Calculus courses. The Root Test is stronger (verifies convergence for more series) than the Ratio Test. This relation inspires introduction of some intermediate strength tests (stronger than the Ratio Test and weaker than the Root Test) that we call Power Mean Tests (they, in particular, include the Arithmetic Mean Test). We show the connection between the Root, the Power Mean and the Ratio Tests. We also note that all these tests are related to the test that we formulate and call Generalized $f$-mean Test (or Kolmogorov-Nagumo-de Finetti mean Test). We provide an example of an infinite series, where the Arithmetic Mean test is the test that should be used for convergence verification, because the Root and the Ratio Tests are not easy to apply. We conclude this work with the statement emphasizing why it is ``fair'' to include the summarizing Corollary 3.1 in our Calculus course.