For all positive integer $k$, let $x_k\in\mathbb{R}$. Assume that $\sum_{k=1}^\infty x_k$ exists. Let $f:\mathbb{R}\to\mathbb{R}$ and assume $\lim_{k\to\infty}f(x_k)=0$. Can we prove or disprove by counterexample that $\sum_{k=1}^\infty f(x_k)$ exists?
I suspect there are obvious counterexamples for this but I am not sure how to find any. Also, I am interested to know if there are conditions on $f$ (e.g., continuous, differentiable, bounded,...) such that $\sum_{k=1}^\infty f(x_k)$ exists. Any help is appreciated.