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July 1, 2025

A Simple Proof to Stirling's Formula

math

Contents

The Stirling's Formula

Theorem. We have

Preliminary Results as Simple Exercises in Mathematical Analysis

Fact 1. Show that for each ,

Fact 2. Show that the function

is decreasing on when .

Fact 3. If is convex, then show that for every with , one has

Fact 4 (Walli's Formula). Define for integer . Prove that for every

Hence by considering , show that

Proof of the Statement

Motivation

By Fact 1 we have

We note that LHS increases to , thus if also converges, we should get at least a nonzero limit. To study this, consider the quotient

Therefore is increasing, however this limit is possibly unbounded.

Next by Fact 2 we can obtain a decreasing sequence by multiplying a factor :

Observe that LHS of is the sams as , so instead we try to study the limit of

Proof to Stirling's Formula

Let be defined as above, now observe that

by Fact 3 we have , therefore a simple computation yields

  • The first inequality in tells us decreases to a limit ;
  • The second inequality in tells us increases to , so .

By using the following form of Walli's formula (modified from Fact 4)

and using we get

Therefore we conclude that

For the ease of memorization this result is usually written as .