mathematical induction a lot of help needed.. work shown

[johnq2k7]

Use mathematical induction to show that:

integral of (x^n)*(e^-x)dx=n! for n>=1

work shown:

n!= (n-3)(n-2)(n-1)n!

limit of e^-x as x--> infinity........ is equal to zero

and the integral of e^-x is........ equal to -e^-x


limit of x^n if n>=1 is equal to infinity


if n=1

the integral of x*e^-x is equal to -(x+1)*e^-x

if n=2

the integral of x^2*e^-x is equal to -(x+1)^2*e^-x

if n=3

the integral of x^3*e^-x is equal to -(x+1)^2*e^-x

therefore is n=infinity

integral of x^infinity*e^-x is equal to -(x+1)^infinity*e^-x

how do I show through mathematical induction that the integral is equal to n! if n>=1 .. i need some help here

 

[galactus]

Are you familiar with the Gamma function. It is defined as:

\(\displaystyle {\Gamma}(n+1)=\int_{0}^{\infty}x^{n}e^{-x}dx\)

Look familiar?.

 

[johnq2k7]

I'm not familar with the gamma function... this function was thought or covered in the textbook I am using.... is there any other method to prove this using "mathematical induction"?