# transformations divergent series

I need help,

If $\forall n\in \mathbb{N}, x_n> 0$ and $\sum_{n=1}^{\infty} x_n$ is divergent then $\sum_{n=1}^{\infty} \frac{x_n}{1+x_n}$ divergent?

thanks for the help

If $\lim_{n\to\infty} x_n=0$, then for large enough $n$ we have $x_n\lt 1$, and therefore $\frac{x_n}{1+x_n} \gt \frac{x_n}{2}$. It follows by comparison that $\sum \frac{x_n}{1+x_n}$ diverges.
If it is not the case that $\lim_{n\to\infty} x_n=0$, then $\frac{x_n}{1+x_n}$ does not have limit $0$, and therefore $\sum \frac{x_n}{1+x_n}$ diverges.