Absolute convergence implies convergence

Absolute convergence implies convergence: If a series $\sum_{n=1}^\infty a_n$ is absolutely convergent , meaning that

converges, then the original series $\sum_{n=1}^\infty a_n$ converges .

This principle is a cornerstone of series theory: it justifies treating absolutely convergent series much like finite sums and underlies results such as invariance under rearrangement .