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Cauchy product (Definition)

Let $a_k$ and $b_k$ 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: \begin{equation} (a \circ b)(k) = \sum_{l=0}^k a_l b_{k-l}. \end{equation}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: \begin{equation} \sum_{k=0}^{\infty} c_k = \left(\sum_{k=0}^{\infty} a_k \right)\cdot \left(\sum_{k=0}^{\infty} b_k \right) = \sum_{k=0}^{\infty} \sum_{l=0}^k a_l b_{k-l}. \end{equation}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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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 5437 times total.

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AMS MSC40-00 (Sequences, series, summability :: General reference works )

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