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Proposition: Cauchy Product of Absolutely Convergent Series

Let $\sum_{n=0}^\infty a_n$ and $\sum_{n=0}^\infty b_n$ be two absolutely convergent series with $A:=\sum_{n=0}^\infty a_n$ and $B:=\sum_{n=0}^\infty b_n$. We define a real sequence $(c_n)_{n\in\mathbb N}$ with

$$c_n:=\sum_{k=0}^n a_{n-k}b_k.$$ Then the series $\sum_{n=0}^\infty c_n$ is also absolutely convergent with $C:=\sum_{n=0}^\infty c_n$, and it is called the Cauchy product of the series $A$ and $B$: $$C=\sum_{n=0}^\infty c_n=\left(\sum_{n=0}^\infty a_n\right)\cdot\left(\sum_{n=0}^\infty b_n\right)=A\cdot B.$$

Table of Contents

Proofs: 1

Mentioned in:

Parts: 1
Proofs: 2
Propositions: 3

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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983