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Chapter 1: Crises in Mathematics: Fourier's Series

2.1 Avoiding Infinite Summations

2.2 Geometric Series

A Question of Definition

> Cauchy's Approach

2.3 Calculating Pi

2.4 The Harmonic Series

2.5 Taylor Series

2.6 Emerging Doubts
Exercises

2.2 Geometric Series (continued)

Cauchy's Approach

Returning to equation (2.2.1), it may be tempting to try to prove it using precisely the associative law that we saw does not work:

(2.2.5)

In 1821, Augustin Louis Cauchy published his Cours d'analyse de l'École Royale Polytechnique (Course in Analysis of the Royal Institute of Technology). One of his intentions in writing this book was to put the study of infinite series on a solid foundation. In his introduction, he writes,

“As for the methods, I have sought to give them all of the rigor that one insists upon in geometry, in such manner as to never have recourse to explanations drawn from algebraic technique. Explanations of this type, however commonly admitted, especially in questions of convergent and divergent series and real quantities that arise from imaginary expressions, cannot be considered, in my opinion, except as heuristics that will sometimes suggest the truth, but which accord little withy the accuracy that is so praised in the mathematical sciences.”

When Cauchy speaks of “algebraic techinque,” he is specifically referring to the kind of technique employed in equation (2.2.5). While this argument is suggestive, we cannot rely upon it.

Cauchy shows how to handle a result such as equation (2.2.1). We need to restrict our argument to the safe territory of finite summations:
(2.2.6)

Cauchy follows the lead of Archimedes. What we call the inifinite series is really just the sequence of values obtained from these finite sums. Approaching the problem in this way, we can see exactly how much the finite geometric series differs from the target value, T = 1/(1–x). The difference is

If we take a value larger than T, is this finite sum eventually below it? If we take a value smaller than T, is this finite sum eventually above it? The target value is the value of if and only if we can make difference as close to 0 as we wish by putting a lower bound on n. This happens precisely when |x| < 1.

Cauchy's careful analysis shows us that equation (2.2.1) needs to carry a restriction:

(2.2.7)

We have stumbled across a curious and important phenomenon. Ordinary equalities do not carry restrictions like this. A statement such as

is valid for any x, as long as the denominator on the right is not 0. Equation (2.2.7) is something very different. It is a statement about successive approximations. The equality does not mean what it usually does. The symbol + no longer means quite the same. Pitfalls abound.

Exercises

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