Why is it possible to frequency sweep with just one sinusoid?
Thanks to geometrikal and Jason R, the sum of exponentials is equal to zero when $\omega_0 = \frac{2\pi}{N} \hat{k}$ and $0 \leq \hat{k} \leq N-1$, which is obvious when you realize that it's just the sum over $2 \hat{k}$ periods of $e^{-j2\omega_0n}$, where $T = \frac{2\pi}{2\omega_0} = \frac{N \pi}{2\pi \hat{k}} = \frac{N}{2 \hat{k}}$.

For $\hat{k} &gt; N-1$, I suppose you would have to zero pad, which is where the digital sinc Jason R. mentions comes in.