Mathematical Analysis Fundamentals

Theorem 3.40

Mertens³⁵

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

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

(3.12)

Proof

Assume that ${\sum }_{n=0}^{\infty }$