Convergence of this summation

Question

Let $f: \mathbb{R} \to \mathbb{C}$ such that for some constant $C \in \mathbb{R}$,

$$|f(t)| \leq \frac{C}{1+t^2}$$

for all $t \in \mathbb{R}$.

Is there anything that can be concluded about the series

$$\sum_{k \in \mathbb{Z}}f(x+2k\pi)$$?