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

We have: $\dfrac{|a_n|}{|1+a_n|}\leq \dfrac{|a_n|}{1-|a_n|}\leq \dfrac{|a_n|}{1-\frac{1}{2}}= 2|a_n| $  for all $n \geq N_0$ since $a_n \to 0$, and using comparison test, the conclusion follows.
