Customer Reviews

Real Analysis: A Historical Approach

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The purpose of this book is straightforward enough: "The need for rigor in analysis is often presented as an end in itself," but here the theory is presented in a manner that "reflects [its] evolution," whence "rigor is ... introduced as an explanation of the convergence of series in general and of the puzzling behavior of trigonometric series in particular." (p. v).

I say that Stahl betrays this cause and ends up doing more harm than good to anyone interested in such an approach to real analysis.

I shall argue for this claim by focusing on the lynchpin of Stahl's enterprise, which is flagged as such already on the first page of the preface: "the nondifferentiability of Euler's Trigonometric Series provides the crucial counterexample that clarifies the need for this careful examination of the foundations of calculus" (p. v).

Very well, let us turn to this "crucial counterexample." It is based on the following three series:

(i) 1/2 = cos(x) - cos(2x) + cos(3x) - cos(4x) + ...

(ii) x/2 = sin(x) - sin(2x)/2 + sin(3x)/3 - sin(4x)/4 + ...

(iii) x^2/4 = pi^2/12 - cos(x) + cos(2x)/2^2 - cos(3x)/3^2 + cos(4x)/4^2 - ...

Now: "The last two are valid for -pi < x < pi ... whereas the first is false for all values of x ... Yet it is the same procedure of differentiating infinite series that leads from the valid last equation to the valid middle equation and from there to the false first equation." (p. 59)

There you have it---the "crucial counterexample" which "clarifies the need" for a "careful examination" of convergence.

Now I will tell you how Euler or any of his contemporaries would have replied to this utter nonsense: they would simply have pointed out that (i) is in fact not "false" at all. When Stahl calls it "false," what he really means is that the partial sums do not converge. Well, yes, Dr. Stahl, of course they don't. We don't need no fancy epsilons to tell us that. Just look at (ii). Since the maximal slope of sin(nx) is n, the maximal slope of sin(nx)/n is 1. In other words, the slopes (i.e., the terms in (i)) are not diminishing with n. So of course you cannot approximate the series by a partial sum. But that does not mean that the series is "false." All it means is that one crude and vulgar method of trying to evaluate it fails. No problem whatever arises from this fact unless one dogmatically insists that this crude and vulgar method is The One and Only Method Acknowledged by Righteous and Pious Men (or "the definition," to use the conventional shorthand for this phrase).

In other words: Stahl's fancy-pants theory of convergence does not solve any problem, but rather creates a problem where previously there was none. Of course this does not "clarify the need" for the theory in question any more than poisoning a well "clarifies the need" for an antidote.