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, $f : X \to R$ is a function, and $g : Y \to X$ is a bijection, then:

[...]

If we consider any bijection $g$ from-to the same set ${i \in Z : n \leq i \leq m}$ , 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.

30.11.2019