This is a question related to a previous question ($a_n$ is an infinite sequence with $\sum\limits_{n=1}^\infty a_n\leq1$ and $0\leq a_n<1$. Prove that $\sum\limits_{n=1}^\infty a_n/(a_n-1)$ converges?).
This time, I want to give a lower bound on the following infinite sum. $a_n$ is an infinite sequence with $\sum\limits_{n=1}^\infty a_n\leq1$ and $0\leq a_n<1$. W.l.o.g., we sort the $a_n$ in descending order, and set $a_1$ equal to $0 < c < 1$. Is it possible to give a lower bound on the sum in terms of $c$ on $\sum\limits_{n=1}^\infty \frac{a_n}{a_n-1}$?
I tried Taylor series, writing the sum in different forms; Jensen's inequaly is the wrong way around...