Cauchy Product of Absolutely Convergent Series

Theorem

Let $\ds \sum_{n \mathop = 0}^\infty a_n$ and $\ds \sum_{n \mathop = 0}^\infty b_n$ be two real series that are absolutely convergent.

Then the Cauchy product of $\ds \sum_{n \mathop = 0}^\infty a_n$ and $\ds \sum_{n \mathop = 0}^\infty b_n$ is absolutely convergent.

Proof

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Sources

• 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2.3$: Operations with series: Theorem $1.7$