# Math Help - kth power of fourier series

1. ## kth power of fourier series

I'm kinda stuck on this problem (attached).. I know it works (from a previous class), but I forgot how to actually prove it.

Thank you!

2. ## Re: kth power of fourier series

If you have:
$S_n^{f}=a_0+\sum_{n=1}^{n} a_n \cos(nx) + \sum_{n=1}^{n} b_n \sin(nx)$

You know the first derivative of:
$a_n \cos(nx) = - a_n \cdot n \cdot \sin(nx)$
and so the second derivative will be
$-a_n \cdot n^2 \cdot \cos(nx)$
...
So the kth derivative will be:
$\pm a_n \cdot n^k \cdot \mbox{trig}(nx)$
We don't know if we get a sinus or a cosinus, that's the reason we wright trig(nx).

Do this also for the other sum.

Does this help you? ...

3. ## Re: kth power of fourier series

thank you! i totally misread the question, and thought it was the series to the kth power, rather than the kth derivative of the series. looks simple enough, problem solved!