Finite geometric series

For $r\ne 1$, a finite geometric series satisfies

Taking the real and imaginary parts of the finite geometric series ${\sum }_{m=1}^N{e}^{im\theta }$ yieldsThese formulas apply when $\theta \notin 2\pi \mathbb{Z}$. At multiples of $2\pi$, the sums are directly $N$ and $0$. Complex exponentials turn two real trigonometric summations into one finite geometric series.

For an interval $I$ of $L$ consecutive integers and $e(x)=e^{2\pi ix}$,where $\Vert \beta \Vert$ is the distance to the nearest integer. This follows from the finite geometric-series formula and $|1-e(\beta )|\asymp \Vert \beta \Vert$.