I am having trouble with deriving a proof for the first difference property for the Fourier Series.
Here is my attempt at the derivation:
$$ y[n] = x[n] - x[n-1] $$
Fourier Series Representation:
$$ a_k - a_ke^{-jk\omega_0} $$
Fourier Series:
$$ y[n] = \sum_{k=<N>}(a_k-a_ke^{-jk\omega_0})e^{jk\omega_0n} $$
I have set up the summation for the Fourier Series, however I have been having difficulty to compute the summation.
The summation should equal this:
$$ a_k(1-e^{-jk\omega_0}) $$
How would I evaluate the summation?