For $x \in \mathbb{R}$ consider the series $F(x)= \sum\limits_{n \in \mathbb{Z} } \exp \left (- \left (x-\frac{n}{2} \right)^2 \right )$. I need to prove that there exists a constant $C$ such that $F(x) \leq C$ for all $x\in \mathbb{R}$.
I have already checked that $F(x) = F(x- \left \lfloor{x}\right \rfloor)$ for all $x \in \mathbb{R}$, so I can reduce the problem to finding an upper bound for all $x \in [0,1]$. And I assume that I need the monotony of $\exp$, but I am not sure how to proceed from here.