I am wondering whether I can get the upper bound in closed form,

$\sum_N(\alpha \exp(2\pi n/N))$ where $\alpha = +1\ or -1$

If alpha is just positive one, this would be just a single value,

but I'm trying to get the upper bound when alpha is +1 or -1, randomly.

I have looked for exponential sums materials, but can't see things like this.