Exploring the Gamma Function: f(3), f(4) and f(5)

[Calcgeek123]

Homework Statement

Define the function: f(x)= The integral from 0 to infinity of t^(x)e^(-t)dt.
Find f(3), f(4) and f(5). Notice anything?

Homework Equations

N/A

The Attempt at a Solution

I assume that I start by finding the integral of f(x). I used wolfram alpha and found that it's apparently the gamma function. I googled the gamma function, but it seems like this crayz thing, and it hasnt gotten me any closer to how to integrate it, especially what f(3), f(4) and f(5) have in common. =/

 

[JG89]

Integrate it using repeated integration by parts. As for the interesting thing you're suppose to see, here's a hint: factorial.

 

[Calcgeek123]

I'm having trouble integrating it because of the t and x. t is a variable, and x is like a number. So when i let u=t^(x), du=xt^(x-1) ..is that correct? Its the x that is throwing me off...

 

[xepma]

First set x equal to 3 (or 4, or 5), and then start solving the integral -- the other way around is actually impossible to solve for general x.

I.e. start by solving this:
$$\int_0^{\infty} t^3e^{-t} dt$$
This gives you f(3). Then the same for x=4, then for x=5, etc. You will need to perform integration by parts -- you don't need a substitution.

 

[Calcgeek123]

That makes sense. Thank you!

I integrated using the tabular method, and got that f(3)=-t^(3)e^(-t)-3t^(2)e^(-t)-6te^(-t)-6e^(-t) from 0 to infinity.

Because one of the bounds includes infinity, I need to take the limit of this function as t goes to infinity. So i did that, and ended up with the limit as t approaches infinity from 0 to infinity of f(3) = (infinity x 0) - (infinity x 0) - (infinity x 0) - 0 -[0-0-0-0]. I'll have to use L'Hospitals rule on the first three terms, but I'm not sure about the [0-0-0-0] part. Each of these terms, when plugging in 0, come out to be (0 x (1/0)) which (1/0) is undefined. Do I just ignore this and make each term 0? Or is that an important part ff the problem..

 

[Calcgeek123]

I found f(3)=6, f(4)=24, and f(5)=120. So f(x)=x!

I need to also determine f'(x). I found f'(x)=t^(3)e^(-t). The next part of the problem says, what does this say about l'hospital's rule and factorials? I know that if i used l'hospitals rule, Id say that f(x)/g(x) = t^3/e^-t. I'm not sure what to do from there though in terms of explaining, or how this relates to factorials. Any suggestions from anyone?