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Math and science::Analysis::Tao::08. Infinite sets

# 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:

$\sum_{x \in X} f(x) := \sum_{i=1}^{n} f(g(i))$

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:

$\sum_{x \in X}f(x) = \sum_{n=0}^{\infty}f(g(n))$

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

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