I can't seem to understand how to derive the "twiddle sum" property:
$$\sum_{n=0}^{N-1}W_{N}^{kn}=N \ \delta[k\bmod N] $$ where $$ W_{N} \triangleq e^{\frac{j 2 \pi }{N}} $$ and $$ \delta[n] \triangleq \begin{cases} 1 & \text{if } n=0 \\ 0 & \text{otherwise} \end{cases}$$
I've tried doing the following:
$$\begin{align} \sum_{n=0}^{N-1}W_{N}^{kn} & = \sum_{n=0}^{N-1}e^{\frac{j 2 \pi k n}{N}} \\ & = \sum_{n=0}^{N-1} \left( e^{\frac{j 2 \pi k}{N}} \right)^n \\ & = \frac{1 - \left( e^{\frac{j 2 \pi k}{N}} \right)^N}{1 - e^{\frac{j 2 \pi k}{N}}} \\ & \\ & = \frac{1 - e^{j 2 \pi k}}{1-e^{\frac{j 2 \pi k}{N}}} \\ & \\ & = \begin{cases} ?? & \text{if } k=mN \quad\quad m\in \mathbb{Z} \\ 0 & \text{otherwise} \quad k \in \mathbb{Z} \end{cases} \\ \end{align} $$
It seems I'm missing an $N$ before the delta, have I done a mistake ?