If a series is convergent is it possible to perform mathematical operations with its sum?

Question

Suppose the series given by $$\left(\displaystyle{\sum_{k=1}^{\infty}|x_{k}|^{2}}\right)^{\frac{1}{2}}$$ is convergent to a number S. Where $x_{k}\in\mathbb{C}$ and $|.|$ is the complex modulus. I want to know if it's possible to know if $\displaystyle{\sum_{k=1}^{\infty}|x_{k}|^{2}}$ is a convergent series. I suppose it is a convergent series since S is a finite number and square a finite number is also finite, am I right?

Answer 1

$S$ is not a series; it is the square root of a series. It turns out you are (in a roundabout way) correct though. $S^2$ is the square of the square root of a series, so it is a series itself.

To answer your general question, yes you can perform arithmetic with a convergent series. Like you said, you can treat it as a finite number.