I have this function which is defined by infinite series
$$f(x)=\sum_{n=1}^{\infty}\frac{1}{x-c_{n}}$$
where $\{c_{n}\}$ is a sequence of nonzero real numbers such that $\sum \frac{1}{c_{n}}<\infty$.
My question is: Is $f$ bounded on $\mathbb R$? i.e. $|f(x)|<\infty$ for all $x\in \mathbb R$?
If not, can we make it bounded by assuming another condition on the sequence $\{c_{n}\}$?