Show the series $a_n/(1+a_n)$ converges absolutely

Question

Given that the series $(a_n)$ converges absolutely. Show that the series $(\frac{a_n}{1 + a_n})$ converges absolutely.

I am not really sure where to start. Any help would be great.

Answer 1

This is true provided that $a_n \neq -1$ for all $n \ge 1$ (so that the fraction is well-defined)

Since $\sum_{n\ge 1} a_n$ converges, $a_n \to 0$, so $\exists$ $N \in \Bbb N$ such that $|a_n| < 1/2$ for all $n > N$. Let $n > N$, then:

$$\left|\frac{a_n}{1 + a_n}\right| = \frac{|a_n|}{|1 + a_n|} \le \frac{|a_n|}{1 - |a_n|} < 2 |a_n|$$

But

$$\sum_{n\ge1} \left|\frac{a_n}{1 + a_n}\right| = \sum_{n=1}^N \left|\frac{a_n}{1 + a_n}\right| + \sum_{n > N} \left|\frac{a_n}{1 + a_n}\right| \le \sum_{n=1}^N \left|\frac{a_n}{1 + a_n}\right| + 2 \sum_{n>N} |a_n| < \infty$$