Math 2.0

Sequences, series, summability Definition

Rearrangement of a series

Builds on Series

Definition

Let ∑n=0∞an\sum_{n=0}^{\infty} a_n be a real series. Let σ ⁣:ℕ→ℕ\sigma \colon \mathbb{N} \to \mathbb{N} be a bijection. The series ∑n=0∞aσ(n)\sum_{n=0}^{\infty} a_{\sigma(n)} is a rearrangement of ∑an\sum a_n.

Remarks

A rearrangement uses each term of the original series exactly once, in a new order. Rearrangements of an absolutely convergent series have the same sum. For a conditionally convergent series, rearrangements can change the sum, by the Riemann rearrangement theorem.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionRearrangement of a series, definition, no evidence levelRearrangement of a se…definition
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  1. Sequence definition
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Further reading

  1. Web https://en.wikipedia.org/wiki/Riemann_series_theorem, Definitions
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