Originally Posted by

**bobcantor1983** I understand how you can do that with addition and that the sum of two convergent series converges, but how can that apply to multiplying two convergent series?

By the absolute value property (and the how the square root is defined), we have that $\displaystyle \left(\frac{|a_n| + |b_n|}{2}\right)*\sqrt{|a_n||b_n|}\geq 0, \forall n$. Following from my previous post, we have that:

$\displaystyle \sqrt{|a_n||b_n|} \leq \frac{|a_n| + |b_n|}{2}, \forall n$.

By the comparison test, $\displaystyle \sum_{n=1}^{\infty}\sqrt{|a_n||b_n|}$ is convergent.