Sequences, series, summability Definition

Cauchy product

Builds on Series

Definition

Let $\sum_{n=0}^{\infty} a_n$ and $\sum_{n=0}^{\infty} b_n$ be real series. For each $n$ , let $c_n = \sum_{k=0}^{n} a_k b_{n-k}$ . The series $\sum_{n=0}^{\infty} c_n$ is the Cauchy product of the two series.

Remarks

The term $c_n$ collects all products $a_i b_j$ with $i + j = n$ . For power series, the Cauchy product of $\sum a_n x^n$ and $\sum b_n x^n$ is $\sum c_n x^n$ . This is the rule for multiplying polynomials. The Cauchy product of two convergent series need not converge. Mertens' theorem gives a sufficient condition.

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Sequence definition
Real sequence definition
Series definition
Cauchy product definition

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47 pages build on this one.

Tag	`0B1K9PVWE9`
URI	`m2://0B1K9PVWE9`
Kind	definition
Writing	W1 stated
MSC	40A05
Defines	Cauchy product
Uses	Series

Cauchy product

Definition

Remarks

Reading path

Where this leads

Further reading

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