PlanetMath: Cauchy product

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Cauchy product

(Definition)

Let and be two sequences of real or complex numbers for $k \in {\mathbb{N}}_0$ ( ${\mathbb{N}}_0$ is the set of natural numbers containing zero). The Cauchy product is defined by:

$\displaystyle (a \circ b)(k) = \sum_{l=0}^k a_l b_{k-l}.$

(1)

This is basically the convolution for two sequences. Therefore the product of two series $\sum_{k=0}^{\infty} a_k$ , $\sum_{k=0}^{\infty} b_k$ is given by:

$\displaystyle \sum_{k=0}^{\infty} c_k = \left(\sum_{k=0}^{\infty} a_k \right)\c... ...\sum_{k=0}^{\infty} b_k \right) = \sum_{k=0}^{\infty} \sum_{l=0}^k a_l b_{k-l}.$

(2)

A sufficient condition for the resulting series $\sum_{k=0}^{\infty} c_k$ to be absolutely convergent is that $\sum_{k=0}^{\infty} a_k$ and $\sum_{k=0}^{\infty} b_k$ both converge absolutely .

"Cauchy product" is owned by msihl.

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Attachments:: multiplication of series (Theorem) by pahio

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Cross-references: converge, absolutely convergent, sufficient, series, product, convolution, natural numbers, complex numbers, real, sequences
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This is version 4 of Cauchy product, born on 2003-05-09, modified 2003-05-20.
Object id is 4254, canonical name is CauchyProduct.
Accessed 4216 times total.

Classification:

AMS MSC:

40-00 (Sequences, series, summability :: General reference works )

Pending Errata and Addenda

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