How can I prove the Laplace transform of t^n step by step?

[Crazy Gnome]

I know the final result, its on all the charts. But I need to show step by step how to get the solution.

If someone could help out by getting me in the right direction or just plain giving me the answer, that would be much appreciated.

 

[nicksauce]

Well probably a best way would be a proof by induction. Do you know what that is?

Edit: I'll add to this. You want to show \int_{0}^{\infty}t^ne^{-st}dt=\frac{n!}{s^{n+1}} for all non-negative integers n. To do this, first show it is true for n=0. Then assume it is true for arbitrary n, and show that it is true for n+1, ie: \int_{0}^{\infty}t^ne^{-st}dt=\frac{n!}{s^{n+1}}\rightarrow\int_{0}^{\infty}t^{n+1}e^{-st}dt=\frac{(n+1)!}{s^{n+2}}. This can easily be done with an integration by parts.

 

[HallsofIvy]

nicksauce, technical point: you should say :"assume it is true for arbitrary n" because that is what you are trying to prove! What you should say is "assume it is true for a single value of n" and then show it is true for n+1. I personally prefer to say "assume it is true for some k" so as not to confuse the specific value with the general value.

Crazy Gnome, the step from n to n+ 1 (or k to k+1) should be easy to do using integration by parts. Oh, and, of course, you need to show it is true for n= 1 or n= 0 depending on whether you are talking about all natural numbers or all whole numbers.

 

[nicksauce]

nicksauce, technical point: you should say :"assume it is true for arbitrary n" because that is what you are trying to prove! What you should say is "assume it is true for a single value of n" and then show it is true for n+1. I personally prefer to say "assume it is true for some k" so as not to confuse the specific value with the general value.

Yes, you're right. I meant to say "an arbitrary n" but left out the "an". And yes, using 'k' is indeed much more clear.