The story of one formula

Abstract

This article aims to represent the diversity of approaches applicable to a certain mathematical problem – Stirling’s approximation was chosen here to achieve the mentioned goal. The first section of the work gives a sight of how the formula appeared, from the derivation of an idea to a publication of the strict results. Further, we provide readers with six different proofs of the approximation. Two of them use methods from calculus and mathematical analysis such that properties of logarithmic function and definite integral as well as representing functions as power series. The other two apply the Gamma function due to its connection with the notion of the factorial, namely Γ(n) = n!, n ∈ N. The last two have a probabilistic idea in their core: both of them combine Poisson distributed random variables with Central Limit Theorem to yield the desired formula. Some of the given proofs are not mathematically rigorous but rather give a sketch of a strict proof. Having all the results we assert that this story can be a good example of the variety of methods that can be used to solve one mathematical problem, even though all the listed proofs use only basic knowledge from several mathematical courses.

Keywords: Stirling’s formula; factorial; Taylor series