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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: [...]