False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passverified 2026-08-10 (gpt-5.6-terra-codex-subscription)

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FALSE: if a convergent series in $\mathbb{R}^{n}$ does not converge absolutely, then every point of $\mathbb{R}^{n}$ is the sum of some rearrangement of it

Statement

False claim: let $n \ge 1$ and let $(x_k)$ be a sequence in $\mathbb{R}^{n}$ whose series converges but does not converge absolutely (Series of vectors in $\mathbb{R}^n$ , absolute convergence, rearrangement, and the set of rearrangement sums). Then every point of $\mathbb{R}^{n}$ is the sum of some rearrangement of $\sum x_k$ ; that is, $\mathcal{S}(x) = \mathbb{R}^{n}$ .

Where the claim comes from. For $n = 1$ it is true, and it is the published 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 $\overline{\mathbb{R}}$ : a conditionally convergent real series can be rearranged to any prescribed sum. The claim above is the naive transfer of that theorem to $\mathbb{R}^{n}$ by analogy, and the analogy fails at $n = 2$ already.

The witness is the series of A convergent series in $\mathbb{R}^{2}$ with $\Gamma$ a line and $\Gamma^{\perp}$ a line, computed from the definition: $x_k = \bigl(\varepsilon_k/\iota(k+1),\ 0\bigr)$ in $\mathbb{R}^{2}$ , which converges, does not converge absolutely, and has no rearrangement sum off the horizontal axis. In particular $(0,1)$ is not a rearrangement sum.

Facts & Assumptions

Given: The sequence $(x_k)$ in $\mathbb{R}^{2}$ of A convergent series in $\mathbb{R}^{2}$ with $\Gamma$ a line and $\Gamma^{\perp}$ a line, computed from the definition, with $x_k = (c_k, 0)$ and $c_k = \varepsilon_k/\iota(k+1)$ .

[A1]

The refuted claim, instantiated at $n = 2$ and this $(x_k)$ : every point of $\mathbb{R}^{2}$ , in particular $(0,1)$ , is the sum of some rearrangement of $\sum x_k$ .

[L1]

The series $\sum x_k$ converges, with sum $s = (S,0)$ , and does not converge absolutely, its norms being $1/\iota(k+1)$ and the harmonic series divergent (A convergent series in $\mathbb{R}^{2}$ with $\Gamma$ a line and $\Gamma^{\perp}$ a line, computed from the definition clauses 1 and 2, For rational $p > 0$ , $\sum 1/k^p$ converges iff $p > 1$ , Absolutely convergent and conditionally convergent series, and the general starting index, The canonical natural $\iota(n) = n \cdot 1_F$ of a field).

[L2]

A rearrangement of $\sum x_k$ is $\sum x_{\sigma(k)}$ for a bijection $\sigma$ of $\mathbb{N}$ , and $\mathcal{S}(x)$ is the set of its sums (Series of vectors in $\mathbb{R}^n$ , absolute convergence, rearrangement, and the set of rearrangement sums, Injection, surjection, bijection).

[L3]

Convergence in $\mathbb{R}^{2}$ is componentwise, partial sums are computed coordinatewise, and a limit in a metric space is unique (For $n \ge 1$ a sequence in $\mathbb{R}^n$ converges iff each coordinate sequence converges, is Cauchy iff each coordinate sequence is Cauchy, and $\mathbb{R}^n$ is complete in every norm clause 1, The standard list $e : n \to F^{n}$ with $e_i(i) = 1_F$ and $e_i(j) = 0_F$ for $j \ne i$ is an ordered basis of $F^{n}$ ; hence $\dim_F F^{n} = n$ , and $F^{0}$ is the zero space with basis $\varnothing$ and dimension $0$ clause 1, A sequence in a metric space has at most one limit, Convergence of a sequence in a metric space: $x_k \to x$ iff $d(x_k, x) \to 0$ in $\mathbb{R}$ , Laws of finite sums and finite products, Finite sums and finite products, by recursion).

[L4]

The containment theorem: $\mathcal{S}(x) \subseteq s + \Gamma^{\perp}$ , and for this series $\Gamma^{\perp}$ is the set of multiples of $e_0$ (The set of rearrangement sums of a convergent series in $\mathbb{R}^n$ is a nonempty subset of the affine subspace $s + \Gamma^{\perp}$ , The subspace $\Gamma$ of directions along which a series converges absolutely, and its orthogonal complement $\Gamma^{\perp}$ , A convergent series in $\mathbb{R}^{2}$ with $\Gamma$ a line and $\Gamma^{\perp}$ a line, computed from the definition clause 3, Linear subspace of a vector space, The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$ , The $p$ -norms $\lVert x\rVert_p$ for rational $p \ge 1$ , and $\lVert x\rVert_\infty$ , A norm on a real vector space, the induced metric, and the dictionary with the metric axioms).

Refutation

technique · direct

1.1

The second coordinate of every term $x_k$ is $0$ , so for every bijection $\sigma$ of $\mathbb{N}$ the second coordinate of every partial sum $\sum_{j<N}x_{\sigma(j)}$ is the finite sum of zeros, namely $0$ .

L2L3

1.2

The hypotheses of the refuted claim are met by this series: it converges and does not converge absolutely.

2.1

If a rearrangement $\sum x_{\sigma(k)}$ converges to a point $t \in \mathbb{R}^{2}$ , then by componentwise convergence its second coordinate sequence, constantly $0$ by step 1.1, converges to $t_1$ ; a constant sequence converges to its value and limits are unique, so $t_1 = 0$ .

step 1.1L3

3.1

Hence every element of $\mathcal{S}(x)$ has second coordinate $0$ , and $(0,1)$ , whose second coordinate is $1 \ne 0$ , is not a rearrangement sum.

step 2.1L2

4.1

So [A1] fails for a series satisfying the hypotheses of the refuted claim, and the claim is false.

step 1.2step 3.1A1

5.1

The failure is structural rather than accidental: by the containment theorem every rearrangement sum lies in $s + \Gamma^{\perp}$ , and here $\Gamma^{\perp}$ is a line in $\mathbb{R}^{2}$ , hence a proper subset, so $\mathcal{S}(x)$ cannot be all of $\mathbb{R}^{2}$ whatever else is true of it.

step 3.1L4∎

Remarks

The refutation uses only the containment half. Step 2.1 is an elementary argument about the second coordinate and needs nothing beyond componentwise convergence; step 5.1 explains it through The set of rearrangement sums of a convergent series in $\mathbb{R}^n$ is a nonempty subset of the affine subspace $s + \Gamma^{\perp}$ , which proves $\mathcal{S}(x) \subseteq s+\Gamma^{\perp}$ and nothing more. No statement about the reverse inclusion is used here, and none is asserted.
Why $n = 1$ is genuinely different. For a conditionally convergent real series $\Gamma = \{0\}$ , so $\Gamma^{\perp}$ is the whole line and the containment says nothing; the space simply has no proper subspace for the rearrangement sums to be trapped in other than $\{0\}$ . From $n = 2$ on there is room, and this witness uses it.
What a correct general statement would have to look like. The affine subspace $s + \Gamma^{\perp}$ is an upper bound for $\mathcal{S}(x)$ , and the two extremes are both realised: it is a single point when the series converges absolutely (An absolutely convergent series in $\mathbb{R}^n$ converges, and every rearrangement converges to the same sum), and it is the whole line in the one-dimensional conditionally convergent case (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 $\overline{\mathbb{R}}$ ). What happens between those extremes for $n \ge 2$ is not settled in this library, and the present item settles only that "everything" is the wrong answer.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 222 results over 44 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Riemann series theorem (Wikipedia) (standard reference, not scraped)
Levy-Steinitz theorem (Wikipedia) (standard reference, not scraped)
T. Banakh, A Simple Inductive Proof of the Levy-Steinitz Theorem (standard reference, not scraped)

FALSE: if a convergent series in Rn\mathbb{R}^{n}Rn does not converge absolutely, then every point of Rn\mathbb{R}^{n}Rn is the sum of some rearrangement of it