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If the series of absolute values converges, then the original series converges.
Absolute convergence implies convergence: If a series is absolutely convergent, meaning that
converges, then the original series 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.
A series is a formal expression together with the associated partial sums .
Whether a series has a value is determined by the behavior of its partial sums: it either converges or diverges. Many standard tools for deciding this are collected as convergence tests such as the ratio test and the comparison test.
An absolutely convergent series is a series such that the series of absolute values is a convergent series.
A convergent series is a series whose partial sums converge to a finite limit , in which case one writes .
Rearrangement theorem (absolute convergence): Let be an absolutely convergent series of real or complex numbers, and let be a bijection. Then the rearranged series
converges, and