Math 2.0

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

Mertens' theorem on Cauchy products

Builds on Cauchy productAbsolute convergenceConvergent series

Statement

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

Remarks

Convergence of both series is not enough. Take an=bn=(−1)n/n+1a_n = b_n = (-1)^n / \sqrt{n + 1}. Both series converge, but ∣cn∣≥1|c_n| \ge 1 for all nn, so the Cauchy product diverges. Franz Mertens proved the theorem.

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 sequencedefinitionCauchy product, definition, no evidence levelCauchy productdefinitionPartial sum, definition, no evidence levelPartial sumdefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionMertens' theorem on Cauchy products, statement, no evidence levelMertens' theorem on C…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. Cauchy product definition
  7. Partial sum definition
  8. Convergent series definition
  9. Absolute convergence definition
  10. Mertens' theorem on Cauchy products statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Cauchy_product, Convergence and Mertens' theorem
About this page
Tag0BJSWZJG18
URIm2://0BJSWZJG18
Kindstatement, theorem
WritingW0 stub
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 0BJSWZJG18.
UsesCauchy product, Absolute convergence, Convergent series

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