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: [ Ссылка ]
►Proof that the Gamma Function gives factorials: [ Ссылка ]
►The Gaussian Integral: [ Ссылка ]
►More reading on Laplace's Method: [ Ссылка ]
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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