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)
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.)