Show that the Gamma function is converging

[crybllrd]

Homework Statement

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

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?

 

[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!