TITLE: How to prove that $\sum_{n=1}^{\infty}\frac{a_{n}}{1+a_{n}}$ converges absolutely
QUESTION [6 upvotes]: If $\sum_{n=1}^{\infty}a_{n}$ converges absolutely, show that
$$\sum_{n=1}^{\infty}\dfrac{a_{n}}{1+a_{n}}$$ converges absolutely.
My try: since $\sum_{n=1}^{\infty}a_{n}$ converges absolutely, then
$$\sum_{n=1}^{\infty}|a_{n}|$$ converges, then there exsit $M>0$, such
$$\sum_{n=1}^{N}|a_{n}|<M$$
then, how to prove than
$$\sum_{n=1}^{N}\dfrac{|a_{n}|}{|1+a_{n}|}<cM?$$
where $c$ is a constant?

REPLY [5 votes]: Hint: Prove that if $\sum a_n$ converges absolutely and $b_n$ is a bounded sequence, then also $\sum a_nb_n$ converges absolutely.

REPLY [4 votes]: In the case that $a_n$ is complex, we notice that
$$\lim_{n\rightarrow\infty}|a_n|=0,$$
so there is $N>0$ such that
$$|a_n|<\frac{1}{2},\quad n\geq N.$$
Then $|1+a_n|\geq 1-|a_n|>\frac{1}{2}$ for $n\geq N$. Hence
$$\sum_{n=0}^{\infty}\frac{|a_n|}{|1+a_n|}=\sum_{n=0}^{N}\frac{|a_n|}{|1+a_n|}+\sum_{n=N+1}^{\infty}\frac{|a_n|}{|1+a_n|}
\leq\sum_{n=0}^{N}\frac{|a_n|}{|1+a_n|}+2\sum_{n=N+1}^{\infty}|a_n|<+\infty.$$
Hence the series converges absolutly.