If $\sum\limits_{n=0}^\infty\frac{a_n}{b_n}$ converges and $\sum\limits_{n=0}^\infty\frac{a_n^2}{b_n^2}$ converges, where $(a_n + b_n)b_n \ne 0$ for every $n \in \mathbb{N}$ , then show that $\sum\limits_{n=0}^\infty \frac{a_n}{a_n + b_n}$ also converges.
I have proved it for non-negative $a_n$ and $b_n$, but I am unable to do it for the other cases.