The (n + 1)th Proof of Stirling's Formula

A review of different approaches to Stirling's formula is given in [4]. A further refer ence is [2], where the purpose is "to give an extremely short proof." There, the author must employ some analysis to justify the application of Lebesgue's dominated conver gence theorem (which, in fact, is not an elementary one). To avoid this application, in [1] we used a skillful but not so obvious estimate of the In function. In the following we present a really elementary deduction of Stirling's formula. Let t > 0, f(x) = xfe~x for x > 0, and A = {x : \x t\ > t/2]. Then