Riemann's Rearrangement Theorem

analysis

Definition

Riemann's Rearrangement Theorem

A conditionally convergent series can be rearranged to converge to any given value $\alpha \in \mathbb{R}$, or to diverge to $+\infty$ or $-\infty$.

Formally, if ${\sum }_{n=1}^{\infty }{a}_n$ is conditionally convergent and $M\in \mathbb{R}\cup \{-\infty ,\infty \}$, there exists a permutation $\sigma$ of $\mathbb{N}$ such that:

$$\sum _{n=1}^{\infty }{a}_{\sigma (n)}=M$$

Visual Intuition

The theorem relies on the fact that the positive and negative terms of a conditionally convergent series both sum to infinity. One can reach any target $M$ by adding positive terms until the sum exceeds $M$, then adding negative terms until it falls below $M$, and repeating this process indefinitely.

Significance

This theorem highlights the fundamental difference between absolute convergence and conditional convergence. In an absolutely convergent series, the sum is invariant under any rearrangement (the commutative property of infinite sums holds). For conditionally convergent series, the order of summation is critical.