Math 2.0

statement theorem

Mertens' theorem on Cauchy products

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.

Explanations

L1 human

One series may converge only conditionally, if the other converges absolutely. Then the product of the two series is still the sum of their Cauchy product.

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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
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  1. Sequence definition
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  3. Limit of a sequence definition
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  6. Cauchy product definition
  7. Partial sum definition
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  9. Absolute convergence definition
  10. Mertens' theorem on Cauchy products statement

Sources

  1. Web https://en.wikipedia.org/wiki/Cauchy_product, Convergence and Mertens' theorem

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