nLab absolute convergence

Definition

For series in the real numbers

In real analysis, given a sequence $a:\mathbb{N} \to \mathbb{R}$ in the real numbers, the series $\sum_{n = 0}^\infty a_n$ is absolutely convergent if the sequence of partial sums is a Cauchy sequence, and the sequence of partial sums of the series $\sum_{n = 0}^\infty \vert a_n \vert$ is also a Cauchy sequence.

For series in Banach spaces

In functional analysis, given a sequence $a:\mathbb{N} \to B$ in a Banach space $B$, the series $\sum_{n = 0}^\infty a_n$ is absolutely convergent if the sequence of partial sums is a Cauchy sequence in $B$, and the sequence of partial sums of the series $\sum_{n = 0}^\infty \Vert a_n \Vert$ is a Cauchy sequence in the real numbers.

See also

• conditional convergence
• Banach space

External links

• Wikipedia, Absolute convergence