I have objects which are labeled by $\phi_{1},...,\phi_{N}$ for some $N > 1$. Is it correct to define their discrete Fourier transforms by: $$\hat{\phi}_{\xi} = \frac{1}{\sqrt{N}}\sum_{k=1}^{N} e^{-ik\xi}\phi_{k}$$ for each $\xi \in \bigg{\{}\frac{2\pi}{N+1}, \frac{4\pi}{N+1},...,\frac{2\pi N}{N+1}\bigg{\}}$ so that its inverse Fourier transform is: $$\phi_{k} = \frac{1}{\sqrt{N}}\sum_{\xi}e^{ik\xi}\hat{\phi}_{\xi}?$$
My doubt concerns the factor $N^{-\frac{1}{2}}$ and also the range (dual lattice) in which $\xi$ is defined. Are my definitions correct? If my definitions are in fact correct, I have to calculate: $$\sum_{k}e^{2ik\xi}$$ for some $\xi$ in the dual lattice. According to my calculations, this gives: $$\sum_{k=1}^{N} e^{2ik\xi} = -1$$ but it looks strange to me. Is this correct?
My reasoning for that is: I'm summing elements of a geometric progression with initial term $e^{2i\xi}$, so this should be: $$\sum_{k=1}^{N} e^{2ik\xi} = \frac{e^{2i\xi}(e^{2i\xi N} - 1)}{e^{2i\xi} - 1} = \frac{e^{2i\xi (N+1)} - e^{2i\xi}}{e^{2i\xi}-1} = \frac{1-e^{2i\xi}}{e^{2i\xi}-1} = -1$$