I need some help figuring out the following proof that my professor put in my notes...
Let . Then, . Prove by induction that where is a polynomial.
This is pretty easy to see why it works as we can continue the product rule forever here and continue to get f(x) times a polynomial. However, I'm not sure how to prove it by induction. Any help would be appreciated.
February 23rd 2011, 07:23 PM
Tis is clearly not true, as in the base step, , and is not a polynomial...
February 23rd 2011, 07:32 PM
Is it true for any n>1?
February 23rd 2011, 08:11 PM
It's false as written for all n>0, but it looks like the following is true:
where is a polynomial.
You should be able to prove this by induction by using the product and quotient rules, and then some algebraic simplification (getting a common denominator). I think you also need to use the fact that the derivative of a polynomial is a polynomial.
February 24th 2011, 02:32 AM
I guess there is no real use in trying to guess the original meaning, but it seems to me that the most reasonable one is that , so that , etc.