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.

**analysis** (differential/integral calculus, functional analysis, topology)

metric space, normed vector space

open ball, open subset, neighbourhood

convergence, limit of a sequence

compactness, sequential compactness

continuous metric space valued function on compact metric space is uniformly continuous

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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 **conditionally convergent** if the sequence of partial sums is a Cauchy sequence, but the sequence of partial sums of the series $\sum_{n = 0}^\infty \vert a_n \vert$ is not Cauchy.

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 **conditionally convergent** if the sequence of partial sums is a Cauchy sequence in $B$, but the sequence of partial sums of the series $\sum_{n = 0}^\infty \Vert a_n \Vert$ is not Cauchy in the real numbers.

- Wikipedia,
*Conditional convergence*

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