From a book on Harmonic Analysis:
$$\large{D_N(\theta) = \sum_{|n|\le N}e^{i n \theta} = e^{-iN\theta} \sum_{0 \le n \le 2N}(e^{i\theta})^n}$$
Evaluating this partial geometric sum, we find that for $\theta \ne 0$,
$$\large{ \bbox[5px,border:2px solid lightgreen]{D_N(\theta) = e^{-iN\theta}\frac{(e^{i\theta})^{2N+1}-1}{e^{i\theta}-1}=\frac{e^{i(2N+1)\theta/2}-e^{-i(2N+1)\theta/2}}{e^{i\theta/2}-e^{-i\theta/2}} }}$$
The boxed green portion is unjustified in the text, and I am having trouble following. What justifies it?
$$ z^{-N} \ \frac{z^{2N+1}- 1}{z - 1} = \frac{z^{N+1}- z^{-N}}{z - 1} = \frac{z^{N+1/2}- z^{-(N + 1/2)}}{z^{1/2} - z^{-1/2}}, $$ where at the last stage, we multiplied top and bottom by $z^{-1/2}$.