Sum of Exponential of i k x

Theorem

 * $\displaystyle \sum_{k \mathop = 0}^n \map \exp {i k x} = \paren {i \sin \frac {n x} 2 + \cos \frac {n x} 2} \frac {\map \sin {\frac {\paren {n + 1} x} 2} } {\sin \frac x 2}$

where $x$ is a complex number that is not an integer multiple of $2 \pi$.