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