Product of Absolutely Convergent Series

Theorem

Let $\map f z = \ds \sum_{n \mathop = 1}^\infty a_n$ and $\map g z = \ds \sum_{n \mathop = 1}^\infty b_n$ be two real or complex series that are absolutely convergent.

Then $\map f z \map g z$ is an absolutely convergent series, and:

$\map f z \map g z = \ds \sum_{n \mathop = 1}^\infty c_n$

where:

$c_n = \ds \sum_{k \mathop = 1}^n a_k b_{n - k}$

Proof

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Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.4$. Power Series