# Riemann's Theorem on Rearranging Conditionally Convergent Series

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

Conditionally convergent series of real numbers have the interesting property that the terms of the series can be rearranged to converge to any real value or diverge to . In this Demonstration, you can select from five conditionally convergent series and you can adjust the target value . The Demonstration rearranges the series, plots its partial sum (the sum from 0 to the term), and shows the rearranged series.

Contributed by: Victor Phan (October 2013)

Suggested by: Simon Tyler

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The five series without rearrangement are

,

,

,

,

,

where is Euler's constant, is the Riemann zeta function, and is the generalized Riemann zeta function.

## Permanent Citation