Please help, I'm stuck...

Prove the equality: int[(t^n)(e^-t)] from 0 to infinity = n!, for n=0,1,2,...

(Evaluate int(e^(-xt)) dt from 0 to infinity and consider whether repeatedly differentiating under the integral is permitted)

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- Jul 28th 2009, 07:25 PMlsutigersintegral
Please help, I'm stuck...

Prove the equality: int[(t^n)(e^-t)] from 0 to infinity = n!, for n=0,1,2,...

(Evaluate int(e^(-xt)) dt from 0 to infinity and consider whether repeatedly differentiating under the integral is permitted) - Jul 28th 2009, 07:52 PMmr fantastic
- Jul 28th 2009, 08:26 PMRandom Variable
I'll give it a try.

We know that

and

so

now let a =1

then - Jul 29th 2009, 06:27 AMHallsofIvy
But where do you show that "repeatedly differentiating under the integral is permitted"?

I think I would be inclined to use "proof by induction", avoiding differentiating under the integral.

If t= 0,

Assume that for some k, [tex]\int_0^\infty t^ke^{-t}dx= k!. Then do using integration by parts: let , so that [tex]du= (k+1)t^k dt[tex] and . Then the integral becomes [tex]= (k+1)k!= (k+1)!

(Ahhh! I see that mr. fantastic already suggested that.) - Jul 29th 2009, 08:06 AMRandom VariableQuote:

But where do you show that "repeatedly differentiating under the integral is permitted"?