Stirling's Incredible Approximation // Gamma Functions, Gaussians, and Laplace's Method
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
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![Factorial's Asymptotic Expansion - DERIVING STIRLING'S FORMULA! [ +Desmos Insights ]](https://i.ytimg.com/vi/-k7s_nXGuO0/hqdefault.jpg?sqp=-oaymwEjCNACELwBSFryq4qpAxUIARUAAAAAGAElAADIQj0AgKJDeAE=&rs=AOn4CLDhRG9Uzo0Us_5V25vGj8H0M_8Gew)
Factorial's Asymptotic Expansion - DERIVING STIRLING'S FORMULA! [ +Desmos Insights ]
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