I'm working on a problem about the convergence of series. As such, I would like to ask for inequalities that relate $\sum_{k=1}^n x_k/(x_k+1)$ and $\sum_{k=1}^n x_k$ where $x_k \ge 0$ for all $k$. My desired inequality is of the form $$\sum_{k=1}^n x_k \le f$$
where $f$ is a function depends on $\sum_{k=1}^n x_k/(x_k+1)$ and $\sum_{k=1}^n x_k$, and $f$ is independent of $n$.
Thank you so much!
