The factorial function: Stirling’s formula

Abstract

How big is n!? For example, to pluck a number out of thin air, what is the order of magnitude of 272! ?This is the third note of a series on the factorial. The two previous notes [1] and [2] dealt with the factorial function x! while here we only consider n! when n is an integer, but its main results also apply to non-integer arguments. This note is longer and more elaborate than the previous ones, but the principal result is in the opening section. Only aficionados need go any further.