Fourier series coefficients: proof by induction

[ElijahRockers]

Homework Statement

Given f = a₀ + sum(a_ncos(nx) + b_nsin(nx))
and f' = a₀' + sum(a_n'cos(nx) + b_n'sin(nx))
The sums are over all positive integers up to n.

show that a₀' = 0, a_n' = nb_n, b_n' = -na_n

Then prove a similar formula for the coefficients of f^(k) using induction.

Homework Equations

The Attempt at a Solution

OK I showed the first part (for the first derivative coefficients) by taking the derivative of f, that was easy enough. I have to confess that the teacher's question also said HINT: (Integrate by parts) but it was much simpler to just take the derivative of f.

I don't know if that's part of the reason I'm having trouble with the induction part or not, but it seems inefficient to integrate backward when it's such a simple thing to show these coefficients with the derivative. I have not had a lot of experience with induction, and I haven't had to do much as an engineering (as opposed to math) major.

So induction works like this: prove it for a single case, assume something is true for some integer k, then use that assumption to show it is also true for k+1. (This is to the best of my understanding)

After considering the problem for some time, i don't think it's too much to show that for any k, a₀^(k) = 0, since the derivative of 0 is 0, and I've shown that the derivative of a₀ = 0.

It's the other coefficients that trip me up, because every time the derivative is taken, the coefficients will take on a value related to its opposite (in terms of functionally odd or even, i mean) coefficient. For example a_n' = nb_n, notice that the derivative a_n' relies on the value of b_n.

I guess I'm just not sure how to start. I feel like I could just do the derivative a bunch of times and look for a pattern, but I feel like that defeats the purpose of proof by induction.

EDIT:
Ok, I wrote out f^(k), then took the derivative and got that a_n^(k+1) = nb_n^(k), but this isn't actually true, because, I did the bunch of derivatives and found that the pattern is actually

a_n^(k+1) = n^kb_n^(k)

How can I show that by induction? I can see that every time the function is differentiated it's going to cause another multiplication by n because of the chain rule, I'm just not quite sure how to show it mathematically.

EDIT2: I integrated f^(k+1) but it didn't seem to really add anything new as far as a solution goes... also the HINT was to integrate by parts but looking at the expression, I don't really see anything to integrate by parts even if i wanted to.

 

[vela]

I think your teacher wants you to calculate the Fourier coefficients for f' using the usual integral formulas for the coefficients. You'll get integrals of the form
$$\int_0^T f' \sin nx\,dx \\ \int_0^T f'\cos nx\,dx $$ which you can evaluate by integrating by parts.