Absolutely Convergent Series

analysis

Definition

Absolutely Convergent Series

An infinite series ${\sum }_{n\ge 0}{a}_n$ is called absolute convergent iff the series of absolute values ${\sum }_{n\ge 0}|a_n|$ is convergent.

An absolute convergent series is also a convergent series. If a series converges but does not converge absolutely, it is called conditionally convergent.

Visual Comparison

The following diagram illustrates a series and its corresponding series of absolute values.

Properties

• Commutativity: The sum of an absolutely convergent series is invariant under any permutation of its terms (unlike conditionally convergent series, see Riemann’s Rearrangement Theorem).
• Completeness: In a Banach space, every absolutely convergent series is convergent.
• Product: The Cauchy product of two absolutely convergent series converges absolutely to the product of their sums.