nLab conditional convergence

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.

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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 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.

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 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.

See also

• absolute convergence
• Banach space
• Riemann series theorem

External links

• Wikipedia, Conditional convergence