Math 2.0

Sequences, series, summability Absolute and conditional convergence, rearrangements, products of series Theorem

Riemann rearrangement theorem

Builds on Conditional convergenceRearrangement of a series

Statement

Let ∑an\sum a_n be a real series that converges conditionally. Let MM be a real number. Then there is a rearrangement ∑aσ(n)\sum a_{\sigma(n)} that converges to MM. There is also a rearrangement that diverges to +∞+\infty, and one that diverges to −∞-\infty.

Remarks

The positive terms and the negative terms each give a divergent series, and an→0a_n \to 0. Take positive terms until the partial sum exceeds MM, then negative terms until it falls below MM, and repeat. So a real series converges absolutely if and only if all its rearrangements converge to the same sum. For the alternating harmonic series, take two positive terms, then one negative term, and repeat. The sum becomes 32ln⁡2\frac{3}{2} \ln 2.

History

Augustin-Louis Cauchy is usually credited with the first remark that a new order can change a sum. He studied the alternating harmonic series in 1833. The theorem is named after Bernhard Riemann.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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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 sumdefinitionRearrangement of a series, definition, no evidence levelRearrangement of a se…definitionConvergent series, definition, no evidence levelConvergent seriesdefinitionAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionConditional convergence, definition, no evidence levelConditional convergen…definitionRiemann rearrangement theorem, statement, no evidence levelRiemann rearrangement…statement
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.
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  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. Rearrangement of a series definition
  8. Convergent series definition
  9. Absolute convergence definition
  10. Conditional convergence definition
  11. Riemann rearrangement theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Riemann_series_theorem, Statement of the theorem
About this page
Tag0E7VBYGFF7
URIm2://0E7VBYGFF7
Aliasesriemann-series-theorem
Kindstatement, theorem
WritingW1 stated
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 0E7VBYGFF7.
UsesConditional convergence, Rearrangement of a series
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