|Dec6-10, 12:54 PM||#1|
1. The problem statement, all variables and given/known data
Say I had a problem like this:
Prove that the nth derivative of x*e^(x) is (x+n)*e^(x) for all integer n.
Can I use reverse induction to prove for negative n? For example...
Say I proved it for my base case, n=0. In this case, the proof is trivial.
Then I prove that if the nth derivative is (x+n)e^(x), then the (n+1)th derivative is (x+n+1)e^(x). (I didn't provide the proof because there's a similar homework problem here, and the proof is easy anyway.
Can I then use reverse induction to prove that if the nth derivative is (x+n)e^(x), then the (n-1)th derivative is (x+n-1)e^(x), thus extending this case to negative derivatives (i.e., integrals)?
Am I even making sense?
|Dec6-10, 05:02 PM||#2|
Hmm, I'm not sure if this is correct. You do have to use reverse induction though. But isn't it easier to show "if it holds for -n, then it holds for -n-1". Or is this what you meant?
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