nLab conditional convergence

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This entry is about conditional convergence of series in real analysis and functional analysis. For conditional convergence of spectral sequences in homological algebra and stable homotopy theory, see conditional convergence of spectral sequences.


Contents

Definition

For series in the real numbers

In real analysis, given a sequence a:a:\mathbb{N} \to \mathbb{R} in the real numbers, the series n=0 a n\sum_{n = 0}^\infty a_n is conditionally convergent if the sequence of partial sums is a Cauchy sequence, but the sequence of partial sums of the series n=0 |a n|\sum_{n = 0}^\infty \vert a_n \vert is not Cauchy.

For series in Banach spaces

In functional analysis, given a sequence a:Ba:\mathbb{N} \to B in a Banach space BB, the series n=0 a n\sum_{n = 0}^\infty a_n is conditionally convergent if the sequence of partial sums is a Cauchy sequence in BB, but the sequence of partial sums of the series n=0 a n\sum_{n = 0}^\infty \Vert a_n \Vert is not Cauchy in the real numbers.

See also

Last revised on January 4, 2023 at 19:08:25. See the history of this page for a list of all contributions to it.