# 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 [...] 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 \( f: X \rightarrow \mathbb{N} \) 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 [...] 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 \( g \).