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: [...]