This is only a partial answer, but I think it is substantial enough to warrant posting:
I'm going to ignore the specific instance and say a bit about summing over general countable sets. First, let me give a rather abstract definition. If this is confusing, skip to the end for a more concrete one, but I do suggest trying to make sense of this one first - it will help set up the right intuition, in my opinion.
Suppose $\Xi$ is a countable set and $f:\Xi\rightarrow\mathbb{C}$ is some function (we could also replace $\mathbb{C}$ with something else without affecting the general picture). There is, I claim, a natural way to try to evaluate the expression $$\sum_{\xi\in\Xi}f(\xi),$$ namely by taking a limit of finite approximations. After all, for each finite $F\subseteq\Xi$ the expression $\sum_{\xi\in F}f(\xi)$ makes perfect sense, and for that matter that's basically how we thought about infinite sums in the "old" setting.
This suggests that we look instead at the expression $$\lim_{F\rightarrow\Xi, F\subseteq_{fin}\Xi}\sum_{\xi\in F}f(\xi).$$ But of course this just pushes us one step back: how do we take the relevant limit? There is, though, a natural way to try to do this: emulate the usual definition of limit at infinity, and say that we get "arbitrarily close" to the desired value as we look at "sufficiently big" finite subsets.
Formally, we say that $$\sum_{\xi\in\Xi}f(\xi)=L$$ iff for all $\epsilon>0$ there is some finite $X\subseteq \Xi$ such that for all finite $Y\subseteq \Xi$ with $X\subseteq Y$ we have $$\vert L-\sum_{\xi\in Y}f(\xi)\vert<\epsilon.$$
(At this point look back at the definition of $\lim_{n\rightarrow\infty}$.) When we say that $\sum_{\xi\in\Xi}f(\xi)$ converges absolutely, we mean that such an $L$ exists.
There is also a more concrete take on things. If $\Xi$ is a countable set, we'll say that an enumeration of $\Xi$ is just an explicit ordering $(\xi_i)_{i\in\mathbb{N}}$ of the elements of $\Xi$ - formally, an enumeration of $\Xi$ is a bijection $\beta:\mathbb{N}\rightarrow\Psi$.
Each enumeration $\beta=(\xi_i)_{i\in\mathbb{N}}$ gives rise to an associate "normal" infinite sum $$Sum(\beta):=\sum_{i\in\mathbb{N}}f(\xi_i)=\lim_{n\rightarrow\infty}\sum_{i=1}^nf(\xi_i).$$
When our countable set starts out with a "default" enumeration, this puts us in familiar territory: changing the enumeration is just rearranging the terms. At this point it's a good exercise to prove:
$\sum_{\xi\in\Xi}f(\xi)$ converges absolutely in the sense of the previous section iff for any two enumerations $\beta,\beta'$ of $\Xi$ we have $Sum(\beta)=Sum(\beta')$.
