Stirling's Incredible Approximation // Gamma Functions, Gaussians, and Laplace's Method

We prove Stirling's Formula that approximates n! using Laplace's Method.
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In his video we prove Stirling's Approximation Formula that n! is approximately sqrt(2 pi n)(n/e)^n in the limit as n goes to infinity. Our proof has three main ingredients. Firstly, we use the Gamma Function definition of n! being a specific improper integral. We will manipulate this integral and apply Taylor's Approximation to turn it into a quadratic, a trick called Laplace's Method. Finally this will become a Gaussian integral and with a change of variables we will get Stirling's Formula.

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Related Info:
►My video comparing n! and n^n:    • The Hierarchy of Big Functions || n^n grea...  
►Proof that the Gamma Function gives factorials:    • Intro to the Laplace Transform & Three Exa...  
►The Gaussian Integral:    • The Gaussian Integral  //  Solved Using Po...  
►More reading on Laplace's Method: http://www.math.utah.edu/~davar/trans...

0:00 Stirling's Formula
0:55 Stirling's Formula graphically
2:25 The Gamma Function
3:09 Taylor Approximations
5:01 The Gaussian Integral
5:33 Laplace's Method

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