Math 2.0

Sequences, series, summability Theorem

Rearrangements of an absolutely convergent series have the same sum

Builds on Absolute convergenceRearrangement of a series

Statement

Let ∑an\sum a_n be a real series that converges absolutely, with sum AA. Let σ ⁣:ℕ→ℕ\sigma \colon \mathbb{N} \to \mathbb{N} be a bijection. Then the rearrangement ∑aσ(n)\sum a_{\sigma(n)} converges absolutely, and its sum is AA.

Remarks

Given ε>0\varepsilon > 0, choose NN with ∑n>N∣an∣<ε\sum_{n > N} |a_n| < \varepsilon. For large mm, the first mm terms of the rearrangement include a0,…,aNa_0, \dots, a_N. So the two partial sums differ by less than ε\varepsilon. By the Riemann rearrangement theorem, the converse also holds for real series. Mathlib states a general form for sums over a type, as Equiv.tsum_eq.

Reading path

To understand this page from the ground up, read these 10 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 convergencedefinitionRearrangements of an absolutely convergent series have the same sum, statement, no evidence levelRearrangements of an …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.
  • 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. Rearrangement of a series definition
  8. Convergent series definition
  9. Absolute convergence definition
  10. Rearrangements of an absolutely convergent series have the same sum statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Absolute_convergence, Rearrangements and unconditional convergence
About this page
Tag0J76TP9FGD
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Kindstatement, theorem
WritingW0 stub
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 0J76TP9FGD.
UsesAbsolute convergence, Rearrangement of a series

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