Prove that for any positive integer ,
Recall the weighted mean value theorem:
For functions and continuous on , if never changes sign in then there exists such that
Proof. In order to apply the weighted mean value theorem (Theorem 3.16 of Apostol) we need to identify the functions and of the theorem. So, let
Furthermore, for a positive integer , does not change sign on the interval (since and do not change sign on this interval). So we may apply the theorem to conclude there exists such that
Now we need to evaluate this integral. (Here I’m going to cheat a little. In the book we do not yet have techniques available to evaluate this integral. We’ll develop the Fundamental Theorem of Calculus, etc, in Chapter 5 that will allow us to do this. If you have a way to do this without directly evaluating an integral that we don’t yet know how to evaluate please do leave a comment with your solution.)
Where the final equality follows since if is even and equals if is odd. Then multiplying that by the , we get .
Putting this together we then have,