# Show that the Gamma function is converging

1. Feb 23, 2012

### crybllrd

1. The problem statement, all variables and given/known data

The gamma function, which plays an important role in advanced applications, is defined for

$n\geq1$ by $\Gamma(n)=\int_0^{\infty} t^{n-1}e^{-t}dt$

(a) Show that the integral converges on $n\geq1$

(b) Show that $\Gamma(n+1)=n\Gamma(n)$

(c) Show that $\Gamma(n+1)=n! if n\geq1$ is an integer

3. The attempt at a solution

I am only having trouble with part (a)

(a) I used integration by parts to get

$e^{-t}nt^{n}+\int_0^{\infty} t^{n}e^{-t}dt$

I also tried it with a different u and dv:

$-t^{n-1}e^{-t}+(n-1)\int_0^{\infty} t^{n-2}e^{-t}dt$

I used this second IBP to find part (b), which I used for (c).

How can I show it is converging?

Last edited: Feb 23, 2012
2. Feb 23, 2012

### crybllrd

Never mind, I figured it out with a comparison test.
Funny, after I post things on here I always seem to figure them out on my own.
Well, whatever works!