TITLE: If $\sum a_n $ is a positive series that diverges, does $\sum \frac{a_n}{1+a_n}$ diverge?
QUESTION [5 upvotes]: Let $ \{a_n\}_{n=1}^\infty $ be a sequence such that $\displaystyle \sum a_n $ that is divergent to $+\infty$.
What can be said about the convergence of $\displaystyle \sum \frac{a_n}{1 + a_n} $?
Any hints, thoughts or leads would be greatly appreciated.
Thanks!

REPLY [4 votes]: If $0 < a_n/(1+a_n) < 1/2$ then $a_n < 1$, and $0 < a_n < 2 (a_n/(1+a_n))$.
So if $\sum a_n/(1+a_n)$ converges, so does $\sum a_n$.