# Riemann's Rearrangement Theorem

## 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.

## 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.