Show : $\sum\limits_{n=1}^\infty \frac{a_n}{1+a_n}$ is absolutely convergent

Question

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

Answer 1

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.

Answer 2

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.