Riemann's Rearrangement Theorem

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Theorem

Let $S$ be an infinite series which is conditionally convergent.

Then its terms can be arranged in a permutation so that the new series converges to any given value, or diverges.


Proof




Also known as

The Riemann's rearrangement theorem is also known as the Riemann series theorem.


Source of Name

This entry was named for Bernhard Riemann.


Historical Note

Riemann's Rearrangement Theorem was an incidental result proved by Bernhard Riemann in his paper Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe of $1854$, on the subject of Fourier series.


Sources