Fourier Series for DT signals
Let $ x[n]\, $ be a periodic DT signal with fundamental period N.
Then $ x[n]=\sum_{k=0}^{N-1} a_k e^{jk\frac{2\pi}{N} n} $
where $ a_k=\frac{1}{N}\sum_{n=0}^{N-1} x_n e^{-jk\frac{2\pi}{N} n} $
note that $ \frac{2\pi}{N} =\omega_0 $
Now consider the signal $ x[n]=sin(3 \pi)\, $
It's periodic because $ \frac{\omega_0}{2\pi} = \frac{\3\pi}{2\pi} =1.5 $is a rational number.
