Show that if $a_n,b_n\in\mathbb{R}$, $(a_n+b_n)b_n\neq0$ and both $\displaystyle\sum_{n=1}^{\infty}\frac{a_n}{b_n}$ and $\displaystyle\sum_{n=1}^{\infty}\left(\frac{a_n}{b_n}\right)^2$ converge, then $\displaystyle\sum_{n=1}^{\infty}\frac{a_n}{a_n+b_n}$ converges.
If $a_n$ is positive, I have been able to solve. How we can solve in general?
from Hot Weekly Questions - Mathematics Stack Exchange
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