Show : $\sum\limits_{n=1}^\infty \frac{a_n}{1+a_n}$ is absolutely convergent Given $\displaystyle \sum_{n=1}^\infty a_n$ is absolutely convergent. Show that $\displaystyle \sum_{n=1}^\infty \dfrac{a_n}{1+a_n}$ also converges absolutely. (If $a_n \neq -1, \forall n \geq 1$ )
I am not sure where to start with this. Any help would be appreciated

From the fact that $\sum_n |a_n|$ converges, one has that $\lim_{n \rightarrow \infty} a_n = 0$. Then you can use the limit comparison test, comparing ${\displaystyle \sum_n \bigg|{a_n \over 1 + a_n}\bigg|}$ to $\sum_n |a_n|$, to show that your series is absolutely convergent, and therefore convergent.
