Rearrangement of a series along a bijection of $\mathbb{N}$, and unconditional convergence

Let $(a_k)$ be a sequence of reals (Sequences of reals: bounded, eventually, frequently, tails, subsequences) and let $\sigma : \mathbb{N} \to \mathbb{N}$ be a bijection (Injection, surjection, bijection).

Rearrangement. The rearrangement of $(a_k)$ along $\sigma$ is the composite sequence $k \mapsto a_{\sigma(k)}$, again a function $\mathbb{N} \to \mathbb{R}$ and so again a sequence of reals. The rearrangement of the series $\sum a_k$ along $\sigma$ is the series $\sum a_{\sigma(k)}$ of that sequence (Series, partial sums, convergence and the sum, divergence, and the tail series).

A rearrangement uses each term of the original sequence exactly once: injectivity of $\sigma$ says no term is repeated, surjectivity says none is omitted. That is the whole content of the word, and it is why the definition is stated with a bijection rather than with an informal "reordering".

Unconditional convergence. The series $\sum a_k$ converges unconditionally when it converges and, for every bijection $\sigma : \mathbb{N} \to \mathbb{N}$, the rearranged series $\sum a_{\sigma(k)}$ converges with

$\sum_{k=0}^{\infty} a_{\sigma(k)} \;=\; \sum_{k=0}^{\infty} a_k .$

• Unconditional convergence implies convergence, by definition and also by instance. The identity map is a bijection of $\mathbb{N}$ and the rearrangement along it is the original sequence, so the clause about all bijections already contains the clause about the series itself.

• Rearranging twice is rearranging once. If $\sigma$ and $\tau$ are bijections of $\mathbb{N}$ then so is $\sigma \circ \tau$, and the rearrangement of $(a_{\sigma(k)})$ along $\tau$ is $k \mapsto a_{\sigma(\tau(k))}$, the rearrangement of $(a_k)$ along $\sigma \circ \tau$. Likewise the inverse $\sigma^{-1}$ is a bijection, and rearranging along it undoes $\sigma$. Both facts are used in Dirichlet's rearrangement theorem: an absolutely convergent series converges unconditionally, and every rearrangement of it has the same sum.

• A weaker-looking condition, which turns out to be the same one. One could ask only that every rearrangement converge, without requiring the sums to agree. Over $\mathbb{R}$ that is not weaker: For a series of real numbers, unconditional convergence and absolute convergence are the same property identifies both conditions with absolute convergence (Absolutely convergent and conditionally convergent series, and the general starting index), because The Riemann series theorem: a conditionally convergent real series has, for every $c \in \mathbb{R}$, a rearrangement with sum $c$, and rearrangements diverging to $+\infty$, to $-\infty$, and oscillating with any prescribed $\liminf \le \limsup$ in $\overline{\mathbb{R}}$ produces, for a series that converges but not absolutely, both a rearrangement with a different sum and a rearrangement that does not converge at all.

• The definition says nothing about which series have the property. That is the subject of the two theorems that follow, and the answer over $\mathbb{R}$ is exactly the absolutely convergent series.