Problem - 4555
Prove the absolute convergence testing rule using the comparison testing rule. That is, if a series $\{|a_n|\}$ converges, then the series $\{a_n\}$ must be convergent.
Because
$$0 \le \sum_{n=1}^{\infty}\left(a_n + |a_n|\right) \le 2\sum_{n=1}^{\infty}|a_n|$$
therefore by the comparison rule we find the series $\{a_n + |a_n|\}$ must converge. It follows that the sum of two convergent series
$$\sum_{n=1}^{\infty}\left(a_n + |a_n|\right) + \sum_{n=1}^{\infty}\left(-|a_n|\right)=\sum_{n=1}^{\infty}\left(a_n\right)$$
must converge.