Stirling's approximation was invented by the Scottish mathematician James Stirling (1692-1770).
using Euler-MacLaurin formula one has
where B1 = −1/2, B2 = 1/6, B3 = 0, B4 = −1/30, B5 = 0, B6 = 1/42, B7 = 0, B8 = −1/30, ... are the Bernoulli numbers, and R is an error term which is normally small for suitable values of p.
Then, for large N,
after some further manipulation one arrives at
|N||N! (exact)||N! (Stirling)||Error (%)|
When one is dealing with numbers of the order of the Avogadro constant () this formula is essentially exact. In computer simulations the number of atoms or molecules (N) is invariably greater than 100; for N=100 the percentage error is approximately 0.083%.