Riemann's Rearrangement Theorem

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

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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 ($\text {1826}$ – $\text {1866}$)