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

Описание: We prove Stirling's Formula that approximates n! using Laplace's Method.
►Get my favorite, free calculator app for your phone or tablet: MAPLE CALCULATOR: https://www.maplesoft.com/products/ma...
►Check out MAPLE LEARN for your browser to make beautiful graphs and much more: https://www.maplesoft.com/products/le...
►Check out Maplesoft's YouTube channel:    / @maplesoft  

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.

Check out my MATH MERCH line in collaboration with Beautiful Equations
►https://www.beautifulequation.com/pag...

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

COURSE PLAYLISTS:
►DISCRETE MATH:    • Discrete Math (Full Course: Sets, Logic, P...  
►LINEAR ALGEBRA:    • Linear Algebra (Full Course)  
►CALCULUS I:    • Calculus I (Limits, Derivative, Integrals)...  
► CALCULUS II:    • Calculus II (Integration Methods, Series, ...  
►MULTIVARIABLE CALCULUS (Calc III):    • Calculus III: Multivariable Calculus (Vect...  
►VECTOR CALCULUS (Calc IV)    • Calculus IV: Vector Calculus (Line Integra...  
►DIFFERENTIAL EQUATIONS:    • Ordinary Differential Equations (ODEs)  
►LAPLACE TRANSFORM:    • Laplace Transforms and Solving ODEs  
►GAME THEORY:    • Game Theory  

OTHER PLAYLISTS:
► Learning Math Series
   • 5 Tips To Make Math Practice Problems Actu...  
►Cool Math Series:
   • Cool Math Series  

BECOME A MEMBER:
►Join:    / @drtrefor  

MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett

SOCIALS:
►Twitter (math based):   / treforbazett  
►Instagram (photography based):   / treforphotography