Real Analysis convergence help I want to show that  $\sum \frac{x_{k}}{(1+ x_{k})}$ diverges if $\sum x_{k}$ diverges. I am asked to consider separately where the sequence is bounded and when it is not.
For the bounded case I believe that I have been able to show converges by the Comparison Test.
However, I am stuck at the unbounded case. So far, all I know is that the sequence is less than one, and each term is close to but less than one so it shouldn't converge.
Thanks!
Thanks everyone for your answers! I get it now!

In the unbounded case, $x_k/(1+x_k)$ becomes close to $1$ and hence bounded away from zero, for infinitely many $k$. Thus the series cannot converge because the terms do not go to zero. The hard case is when $\{x_k\}$ is bounded.
