real analysis - Absolutely convergent series of complex numbers

If $\sum_{n=0}^{\infty} a_n$ is an absolutely convergent series of complex numbers and $a_n \ne -1$ $\forall$ $n$, prove that the series

$\sum_{n=0}^{\infty} \frac{a_n}{1+a_n}$ is absolutely convergent.

I'm not sure about this series in $\Bbb C$... I haven't worked with complex series before so I'm not quite sure how to prove this. I'm trying to start with a proof in $\Bbb R$, but I'm not getting very far. Placing a series in a series is tough to wrap my mind around when the only thing I know is that $a_n \ne -1$. Which is an obvious fact since then the fraction would be undefined.