# Riemann's Rearrangement Theorem

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## Theorem

Let $S$ be a real 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

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## 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

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.32$: Riemann ($\text {1826}$ – $\text {1866}$)