Projects & Research A Radical Approach to Real Analysis Macalester College


Chapter 1: Crises in Mathematics: Fourier's Series

Chapter 2: Infinite Summations

2.1 Avoiding Infinite Summations

2.2 The Geometric Series

2.3 Calculating Pi

2.4 The Harmonic Series

2.5 Taylor Series

2.6 Emerging Doubts


Chapter 2: Infinite Summations

The term infinite summation is an oxymoron. Infinite means without limit, non terminating, never ending. Summation is the act of coming to the highest point (summus, summit), reaching the totality, achieving the conclusion. How can we conclude a process that never ends? The phrase itself should be a red flag alerting us to the fact that something very subtle and non intuitive is going on. It is safer to speak of an infinite series for a summation that has no end, but we shall use the symbols of addition, the + and the . We need to remember that they no longer mean quite the same thing.

In this chapter, we will see why infinite series are important. We will also see some of the ways in which they can behave totally unlike finite summations. The discovery of Fourier series accelerated this recognition of the strange behavior of infinite series. We will learn more about why they were so disturbing to mathematicians of the early 19th century.

We begin by learning how Archimedes of Syracuse dealt with infinite series. In some respects, he was more careful than needed to be . But, ultimately, his approach became the one that mathematicians would adopt.

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