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单词 CauchyProduct
释义

Cauchy product


Let ak and bk be two sequencesPlanetmathPlanetmath of real or complex numbers fork0 ( 0 is the set of natural numbers containing zero).The Cauchy productMathworldPlanetmath is defined by:

(ab)(k)=l=0kalbk-l.(1)

This is basically the convolution for two sequences.Therefore the product of two series k=0ak, k=0bk is given by:

k=0ck=(k=0ak)(k=0bk)=k=0l=0kalbk-l.(2)

A sufficient condition for the resulting series k=0ck to be absolutely convergent is that k=0ak and k=0bk both converge absolutely .

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更新时间:2025/5/4 10:57:23