Math 2.0

Sequences, series, summability Theorem

Cauchy product of absolutely convergent series

Builds on Cauchy productAbsolute convergence

Statement

Let ∑an\sum a_n and ∑bn\sum b_n be real series that converge absolutely, with sums AA and BB. Then their Cauchy product ∑cn\sum c_n converges absolutely, and its sum is ABA B.

Remarks

The double series ∑i,jaibj\sum_{i,j} a_i b_j converges absolutely, so its terms can be grouped in any order. The Cauchy product groups them by i+ji + j. Mertens' theorem needs absolute convergence of only one of the two series, but then gives only convergence. Mathlib states it for complete normed rings, as tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm.

Reading path

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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 sequencedefinitionCauchy product, definition, no evidence levelCauchy productdefinitionPartial sum, definition, no evidence levelPartial sumdefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionCauchy product of absolutely convergent series, statement, no evidence levelCauchy product of abs…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. Cauchy product definition
  7. Partial sum definition
  8. Convergent series definition
  9. Absolute convergence definition
  10. Cauchy product of absolutely convergent series statement

Where this leads

45 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cauchy_product, Convergence and Mertens' theorem
About this page
Tag0AN28XQ45Z
URIm2://0AN28XQ45Z
Kindstatement, theorem
WritingW0 stub
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 0AN28XQ45Z.
UsesCauchy product, Absolute convergence

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