Product of Absolutely Convergent Series

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Theorem

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


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

$f \paren z g \paren z = \displaystyle \sum_{n \mathop = 1}^\infty c_n$

where:

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


Proof


Sources