Theorem 3.40

Mertens^{35}

*Let* ${\sum}_{n=0}^{\infty}{a}_{n}$ *and* ${\sum}_{m=0}^{\infty}{b}_{m}$ *be convergent numerical series such that at least one of them is absolutely convergent. Then the series* ${\sum}_{n\text{,}m=0}^{\infty}{a}_{n}{b}_{m}$ *converges and*

$\sum _{n\text{,}m=0}^{\infty}}{a}_{n}{b}_{m}=\left({\displaystyle \sum _{n=0}^{\infty}}{a}_{n}\right)\left({\displaystyle \sum _{m=0}^{\infty}}{b}_{m}\right)\text{.$ (3.12)

(3.12)