Mathematics

The key insight is that absolute convergence gives you unconditional control over tail behavior. When $\sum |a_n|$ converges, you can make the sum of any finite subset of terms as small as you like by excluding only finitely many terms—this property survives any rearrangement. A rearrangement just reorders which terms you're adding, but it can't change the fact that you're eventually summing all the same terms, and the absolute convergence guarantee ensures the order doesn't matter for the final limit.

The key insight is that absolute convergence gives you unconditional control over tail behavior. When $\sum |a_n|$ converges, you can make the sum of any finite subset of terms as small as you like by excluding only finitely many terms—this property survives any rearrangement. A rearrangement just reorders which terms you're adding, but it can't change the fact that you're eventually summing all the same terms, and the absolute convergence guarantee ensures the order doesn't matter for the final limit.