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!

Suppose that $\{x_n\}$ is a bounded sequence.  That is, there exists an $M \geq 0$ such that
$|x_n| \leq M$ for all $n$.  Then, $\sum_{n=1}^N \frac{x_n}{1+x_n} \geq \frac{1}{1+M} \sum_{n=1}^N x_n \to \infty$ as $N \to \infty$. 
