# Proof of First Difference Property for Fourier Series

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:

$$y[n] = \sum_{k=}(a_k-a_ke^{-jk\omega_0})e^{jk\omega_0n}$$

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?

You have the result already written down in your question. If $$a_k$$ are the Fourier coefficients of $$x[n]$$, then your third formula is the Fourier series of $$y[n]=x[n]-x[n-1]$$. So the Fourier series coefficients of $$y[n]$$ are
$$b_k=a_k-a_ke^{-jk\omega_0}=a_k(1-e^{-jk\omega_0})$$
$$y[n]=\sum_{k=0}^{N-1}b_ke^{j\omega_0 nk},\qquad\omega_0=2\pi/N$$