34 - The Harmonic Series: A Primer

Summary

Introduction

Students in a first course of real analysis are often bewildered by many things, but perhaps the main difficulties they encounter are centered around three fundamental concepts: the notion of infinity and infinite processes; the phenomena of convergence and divergence; and the construction of rigorous proofs. It is in just such a course that historical information can be used to good effect. After all, the fact that these are issues with which mathematicians have grappled for centuries will doubtless be of some reassurance to the student struggling to master them.

What may be less comforting (but important, nevertheless, for a student to know) is that while mathematical results, once proved, are permanent, the arguments on which they are based are sometimes less so. In other words, as mathematics has developed, so has the concept of mathematical rigor. The result is that, although a theorem first proved 250 years ago is still true, the proof originally given for it might not be regarded as fully satisfactory by today's mathematicians. Given that the result still holds, however, it is not normally too difficult to formulate an alternative proof that attains modern standards of mathematical rigor.

A good example is the subject of this chapter, the harmonic series, which provides both a simple and an important introduction to issues surrounding the convergence and divergence of infinite series. Moreover, by virtue of its rich history and the related mathematical topics that arise from its study, it is a particularly appropriate tool to use when introducing students to the rudiments of classical analysis.