Math 2.0

Sequences, series, summability Definition

Conditional convergence

Builds on Convergent seriesAbsolute convergence

Definition

Let (an)(a_n) be a real sequence. The series ∑an\sum a_n converges conditionally if ∑an\sum a_n converges and ∑∣an∣\sum |a_n| diverges.

Remarks

So a convergent series converges either absolutely or conditionally, and not both. The alternating harmonic series converges conditionally. By the Riemann rearrangement theorem, the terms of a conditionally convergent series can be reordered to give any real sum.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionConditional convergence, definition, no evidence levelConditional convergen…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Sequence definition
  2. Real sequence definition
  3. Limit of a sequence definition
  4. Series definition
  5. Convergent sequence definition
  6. Partial sum definition
  7. Convergent series definition
  8. Absolute convergence definition
  9. Conditional convergence definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Conditional_convergence, Definition
About this page
Tag02YRQXZDCS
URIm2://02YRQXZDCS
Aliasesconditionally-convergent-series
Kinddefinition
WritingW1 stated
MSC40A05
DefinesConditional convergence
UsesConvergent series, Absolute convergence

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