GraphingCalculator 4;
Window 45 14 865 1274;
PaneDivider 415;
FontSizes 18;
StackPanes 1;
2D.BottomLeft -0.53125 -2;
2D.GraphPaper 0;
Text "Stirling's approximation";
Color 4;
Expr M=10;
Color 3;
Expr vector(N,N!),in(N,set(0*ldots*M));
Color 17;
Expr x!;
Color 3;
Expr function(S,x)=x^x*e^(-x)*sqrt(2*pi*x);
Color 17;
Expr function(S,x);
Text "Why does it work?  Elements of two proofs.

First, show ln(n!) is pretty well approximated by the area under ln(x) up to x=n";
Color 17;
Expr ln(x);
Color 17;
Expr 0<y<ln([N]),N-1/2<x<N+1/2,in(N,set(1*ldots*M));
Text "Second, show the integrand defining the gamma function n! is well approximated by a n^n*exp(-n) times a Gaussian:";
Color 17;
Expr x^M*e^(-x);
Color 17;
Expr x^M;
Color 17;
Expr e^(-x);
Color 17;
Expr x=M;
Color 17;
Expr M^M*e^(-M)*e^(-[x-M]^2/(2*M));