The same question in $\mathbb{R}^d$: what the set of rearrangement sums looks like, and why that answer is not reachable at this point in the reading order

Everything on this page is about series of real numbers, and the answer it reaches is complete for that case. Write

$\mathcal{S}(a) \;:=\; \Bigl\{\, s \in \mathbb{R} \ : \ \text{some rearrangement of } \sum a_k \text{ converges to } s \,\Bigr\}$

for the set of rearrangement sums of a convergent series (Rearrangement of a series along a bijection of $\mathbb{N}$, and unconditional convergence). Then this page determines $\mathcal{S}(a)$ exactly, in two cases and no others.

• If $\sum a_k$ converges absolutely, $\mathcal{S}(a)$ is the single point $\bigl\{\sum_{k=0}^{\infty} a_k\bigr\}$: that is Dirichlet's rearrangement theorem: an absolutely convergent series converges unconditionally, and every rearrangement of it has the same sum.
• If $\sum a_k$ converges conditionally, $\mathcal{S}(a)$ is the whole of $\mathbb{R}$: that is 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 $\overline{\mathbb{R}}$, and moreover rearrangements exist whose partial sums diverge to $+\infty$ or to $-\infty$ or oscillate between any prescribed pair of extended reals.

For a series of real numbers, unconditional convergence and absolute convergence are the same property is the statement that these two cases are distinguished by absolute convergence and by nothing else.

The same question can be asked of a series of vectors, once one has a space in which a series of vectors has a sum: given a convergent series in $\mathbb{R}^d$, what does its set of rearrangement sums look like? That question was raised by Paul Lévy in 1905 and taken up by Ernst Steinitz in 1913, and later by Wacław Sierpiński; the references below are to those papers, and they are given as the origin of the question. What the literature answers is not stated here in any form, and nothing on this page or anywhere else in this library depends on it. Part of it is now proved, later in the reading order and marked as forward material: Steinitz's polygonal confinement theorem: finitely many vectors of norm at most $1$ summing to $0$ can be ordered so that every partial sum has norm at most $n$ ↗ and The set of rearrangement sums of a convergent series in $\mathbb{R}^n$ is a nonempty subset of the affine subspace $s + \Gamma^{\perp}$ ↗ establish that the set of rearrangement sums is nonempty and lies inside an affine subspace. The reverse inclusion, which is what would turn that containment into the classical answer, is still proved nowhere here.

The reason is a matter of reading order, not of difficulty or of interest. Stating the theorem requires $\mathbb{R}^d$ as a normed space (a norm, convergence of vector sequences, and a notion of a convergent series of vectors), and that vocabulary is introduced later in the reading order than this page. Rather than borrow it, or state a theorem whose terms are not yet defined, the obligation is recorded where it can be discharged: on the page that builds $\mathbb{R}^d$ as a normed space and afterwards. When that page is reached, the question raised here is the one it will answer.

What is safe to say now, and is worth saying. The one-dimensional dichotomy above is stark: a single point, or everything. Nothing in the proof of 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 $\overline{\mathbb{R}}$ survives verbatim in higher dimensions, because it is built on the order of $\mathbb{R}$: the greedy rule "add positive terms until the running sum exceeds the target, then negative ones until it falls below" presupposes that the terms are signed and that the target can be approached from two sides. In $\mathbb{R}^d$ with $d \ge 2$ there is no such order, the terms point in many directions, and the argument has no analogue. A reader who expects the one-dimensional answer to generalise unchanged should treat that expectation as unsupported until the later page settles it.

No claim of this library is made about $\mathbb{R}^d$ above. The two Lévy and Steinitz papers are cited as the historical source of the question, not as authority for a result used anywhere here; no item on this page or elsewhere in the library rests on them.