Suppose that \(\sum^{\infty}_{n=1} |a_n|\) converges. It is trivial that either \(a_n=|a_n|\) or \(a_n = -|a_n|\) and therefore either \(|a_n|+a_n=2|a_n|\) or \(|an|+an=0\). This means:
\[ 0 ≤ |a_n| + a_n ≤ 2 |a_n|\]
If \(\sum^{\infty}_{n=1} |a_n|\) converges, then so does \(2\sum^{\infty}_{n=1} |a_n|\), which would also mean that \(\sum^{\infty}_{n=1} |a_n| + a_n\) also converges. By using the algebraic properties for convergent series:
\[ \sum^{\infty}_{n=1} a_n = \sum^{\infty}_{n=1} |a_n| + a_n - \sum^{\infty}_{n=1} |a_n|\]
If both \(\sum^{\infty}_{n=1} |a_n| + a_n\) and \(\sum^{\infty}_{n=1} |a_n|\) converge, then so does \(\sum^{\infty}_{n=1} a_n\).