Series on countable sets, definition
Previously, summation over finite sets were defined:
With some bijection \( g: \{i \in \mathbb{N} : 1 \le i \le n\} \rightarrow X \), the sum over \( X \) is defined as:
We can extend this notion to summation over an infinite set \( X \), as long as we have a bijection between \( \mathbb{N} \) and \( X \)—in other words, \( X \) is countable. 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 \( f: X \rightarrow \mathbb{N} \) be a function. We say that the series \( \sum_{x \in X}f(x) \) is absolutely convergent iff for some bijection \( g : \mathbb{N} \rightarrow X \), the sum \( \sum_{n=0}^{\infty}f(g(n)) \) 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 \( g \).