Series on countable sets, definition

Previously, summation over finite sets were defined:

We can extend this notion to summation over an infinite set $X$, as long as we have a [...] between $\mathbb{N}$ and $X$—in other words, $X$ is [...]. In addition, our notion of summation is limited to absolute convergence; we have previously shown that a rearrangement of terms of a conditionally but not absolutely convergent series does not necessarily converge to the original sum.

Series on countable sets

Let $X$ be a countable set, and let $f:X\to \mathbb{N}$ be a function. We say that the series $\sum _{x\in X}f(x)$ is absolutely convergent iff for some bijection $g:\mathbb{N}\to X$, the sum [...] is absolutely convergent. We then define the sum of $\sum _{x\in X}f(x)$ by the formula:

Tao later notes that this definition is sufficient for it to be true for any bijection $g$.