prove series converges

Question

Question: Given sequences $(a_{n})$, $(b_{n})$, $a_{n}(a_{n}+b_{n})\neq 0$ , $n\ge1$ and both $\sum_{n=1}^{\infty} \frac {a_{n}}{b_{n}}$ ,$\sum_{n=1}^{\infty} \frac {a_{n}^2}{b_{n}^2}$ converges. Prove $\sum_{n=1}^{\infty} \frac {a_{n}}{b_{n}+a_{n}}$ converge.

My idea is this let $\frac {a_{n}}{b_{n}}=x_{n}$ then we have both $\sum_{n=1}^{\infty} {x_{n}}$, $\sum_{n=1}^{\infty} {x_{n}^2}$ converges, we need prove $\sum_{n=1}^{\infty} \frac {x_{n}}{1+x_{n}}$ converge. But I have trouble proving it, is it possible?

Answer 1

With your notation write $$ \frac{x_n}{1+x_n}=x_n-\frac{x_n^2}{1+x_n}. $$ Since $\sum x_n$ converges, $\lim_{n\to\infty}x_n=0$. For all $n$ large enough we have $|x_n|\le1/2$, $1+x_n\ge1/2$ and $$ 0<\frac{x_n^2}{1+x_n}\le2\,x_n^2. $$

Answer 2

Hint.

For $\vert x \vert \le 1$ you have $$\left\vert \frac{x}{1+x}-(x-x^2) \right\vert \le \vert x^2 \vert$$