On a heuristic derivation of Stirling's approximation to n!

Abstract

An elementary discussion that appears in at least two texts and purportedly suggests the basis of an argument for the derivation of Stirling's formula is reproduced, together with objections to two steps in the discussion that are essential to the simplification of the argument. These steps are the use of the Maclaurin expansion for In (1 + u) outside its circle of convergence followed by the deletion of certain terms which do not in fact individually tend to zero under the required conditions.

A more suggestive account is developed along the same lines as the first, which provides both alternative explanations for essentially the same simplifications, and points the way to a rigorous though somewhat lengthy derivation of the required result.

Notes

† Now on Secondment to the Faculty of Mathematics, Open University, Milton Keynes, England.

Additional information

Notes on contributors

R. Renner†

† Now on Secondment to the Faculty of Mathematics, Open University, Milton Keynes, England.