How to prove the convergence of a series with absolute values?

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Happy New Year everyone! Here is this week's POTW:

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If $\sum a_n$ is an absolutely convergent series of nonzero real numbers, prove that $\sum \dfrac{a_n^2}{1 + a_n^2}$ converges. If, in addition, $a_n \neq -1$ for all $n$, show that $\sum \dfrac{a_n}{1 + a_n}$ converges absolutely.

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No one answered this week's problem. You can read my solution below.

Spoiler

By the estimate $1 + a_n^2 \ge 2|a_n|$, the series $\sum \dfrac{a_n^2}{1 + a_n^2}$ is dominated by the convergent series $\dfrac{1}{2}\sum |a_n|$. By the comparison test $\sum \dfrac{a_n^2}{1 + a_n^2}$ converges. If $a_n \neq -1$ for all $n$ as well, then using the fact that $|a_n| < 1/2$ for all sufficiently large $n$, it follows that $$\left|\frac{a_n}{1 + a_n}\right| \le \frac{|a_n|}{1 - |a_n|} < 2|a_n|$$ for $n$ sufficiently large. Since $\sum 2 |a_n| < \infty$, by comparison the series $\sum \dfrac{a_n}{1 + a_n}$ converges absolutely.