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Published online by Cambridge University Press: 02 December 2009
Stirling's approximation to the factorial function, n!∼(2πn)½(n/e)n (1) plays a central role in any number of investigations of statistical physics, and is invaluable in the kinds of simple probabilistic studies that can convey to students in a general education course the nature of entropy and irreversibility. Unfortunately, the usual derivations of (1) are inaccessible to such students and even to many beginning physics majors. One can, of course, simply verify its remarkably accurate performance, but the better students are bound to find this frustrating: Why is it that Stirling's formula works as well as it does?
I provide here an elementary answer to this question that can be adapted to give convincing explanations at a range of levels of mathematical innocence. For the crudest argument it is only necessary to know the elementary definition of the number e that arises in the theory of compound interest. Students who also know that the natural logarithm has the expansion
can be given a really intimate glimpse into the workings of Stirling's formula, while those who are willing to approximate a few simple sums by integrals can acquire a level of understanding possessed, I suspect, by few professionals.
Stirling's formula begins to yield up its secrets with the observation that n! can evidently be written in the form
This can be written in the equivalent form,
which immediately calls to mind the definition of e as the limiting value for large.
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