Math and science::Analysis::Tao::07. Series

# Rearrangement of infinite series

A feature of finite series which we will recap here is that any rearrangement of the terms of the series does not affect the sum. For example:

$a_1 + a_2 + a_3 + a_4 = a_4 + a_1 + a_3 + a_2$

This comes from the first property of substitution:

If $$X$$ is a finite set, $$f: X \rightarrow R$$ is a function, and $$g : Y \rightarrow X$$ is a bijection, then:

[...]

If we consider any bijection $$g$$ from-to the same set $$\{ i \in \mathbb{Z} : n \le i \le m \}$$, then we can say:

$\sum_{i=n}^{m} a_i = \sum_{i=n}^{m} a_{g(i)}$

which is the basis for the rearrangement example above.

Can we rearrange the terms of an infinite series and get the same result? Yes and no.

• An absolutely convergent series: [...]
• Conditionally, but not absolutely convergent series: [...]