33 - Amo, Amas, Amat! What's the sum of that?

Summary

Introduction

Every course in undergraduate calculus contains some component of the examination of series and the various tests to establish their convergence. One of the most important series is the Harmonic series, which is not only mathematically interesting per se, but also appears frequently as an ideal ‘comparison’ series to determine the convergence or divergence of other series. At some point, the formal proof of its divergence must be covered. This paper provides a quirky alternative to the format and the content of the standard proof usually offered; a capsule based on an examination of the actual primary source of the proof, as it originally appeared, in Latin.

This capsule should ideally be offered before covering the various convergence tests, and just after examining geometric series. It could be particularly fitting to include it as part of your coverage of the divergence test as the Harmonic series is often the example cited to demonstrate that the convergence of terms in a series that tend to zero is not sufficient to guarantee the convergence of the actual series.

Given the richness of historical insight, the relevance of the mathematics, and indeed the novelty for the students, the presentation of this primary source is ideal for the undergraduate mathematics classroom. Grabbing the attention of the students by presenting something completely different, yet utterly relevant, may very well renew their enthusiasm as well as stimulating curiosity and assisting their grasp of this topic.