ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-29

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A convergent series in $R^{2}$ with $Γ$ a line and $Γ^{⊥}$ a line, computed from the definition

Example

Let $(ε_{k})$ be the alternating sequence, the unique sequence of reals with $ε_{0} = 1$ and $ε_{k + 1} = - ε_{k}$ , so $∣ ε_{k} ∣ = 1$ for every $k$ (The even and odd index maps and the alternating sequence: strictly increasing $e, o$ with $N$ their disjoint union, and the unique $(s_{k})$ with $s_{0} = 1$ , $s_{σ (k)} = - s_{k}$ , which satisfies $∣ s_{k} ∣ = 1$ , $s \circ e \equiv 1$ and $s \circ o \equiv - 1$ ). In $R^{2}$ put

$x_{k} : = (\frac{ε_{k}}{ι (k + 1)}, 0) (k \in N),$

with $ι$ the canonical natural (The canonical natural $ι (n) = n \cdot 1_{F}$ of a field). Call a line through the origin the set of scalar multiples of a fixed nonzero vector; each such set is a linear subspace (Linear subspace of a vector space). Then:

$\sum x_{k}$ converges, to $s = (S, 0)$ where $S$ is the sum of the alternating harmonic series; the value of $S$ is not computed here, being a logarithm and outside this page's reach.
$\sum x_{k}$ does not converge absolutely (Series of vectors in $R^{n}$ , absolute convergence, rearrangement, and the set of rearrangement sums).
$Γ = {(0, t) : t \in R}$ , the line of multiples of $e_{1}$ , and $Γ^{⊥} = {(t, 0) : t \in R}$ , the line of multiples of $e_{0}$ (The subspace $Γ$ of directions along which a series converges absolutely, and its orthogonal complement $Γ^{⊥}$ ).
Consequently The set of rearrangement sums of a convergent series in $R^{n}$ is a nonempty subset of the affine subspace $s + Γ^{⊥}$ confines every rearrangement sum to the horizontal line $s + Γ^{⊥} = {(t, 0) : t \in R}$ ; and for this series the confinement is exact, $S (x) = s + Γ^{⊥}$ , by the published The Riemann series theorem: a conditionally convergent real series has, for every $c \in R$ , a rearrangement with sum $c$ , and rearrangements diverging to $+ \infty$ , to $- \infty$ , and oscillating with any prescribed $lim inf \leq lim sup$ in $\overline{R}$ applied to the first coordinate.

Clause 4 decides nothing about the general question. This series is degenerate: it lies inside a line, so its rearrangement behaviour is the one-dimensional behaviour of its first coordinate and nothing more. It is therefore not evidence about whether $S (x) = s + Γ^{⊥}$ for a series genuinely spread over $R^{n}$ with $n \geq 2$ , a question this library does not settle (Conventions of this page, the standing $n \geq 1$ hypothesis, and what is taken up elsewhere in the reading order).

Facts & Assumptions

Given: The sequence $(x_{k})$ above, its first coordinate sequence $c_{k} : = ε_{k} / ι (k + 1)$ and the sequence $b_{k} : = 1 / ι (k + 1)$ .

[L1]

The alternating sequence: $ε_{0} = 1$ , $ε_{k + 1} = - ε_{k}$ , and $∣ ε_{k} ∣ = 1$ (The even and odd index maps and the alternating sequence: strictly increasing $e, o$ with $N$ their disjoint union, and the unique $(s_{k})$ with $s_{0} = 1$ , $s_{σ (k)} = - s_{k}$ , which satisfies $∣ s_{k} ∣ = 1$ , $s \circ e \equiv 1$ and $s \circ o \equiv - 1$ , Basic properties of the absolute value).

[L2]

$ι (k + 1) > 0$ and $ι$ is strictly increasing; $0 < u \leq v$ gives $0 < 1 / v \leq 1 / u$ ; and for every real $ε > 0$ there is $K$ with $1 / ι (K + 1) < ε$ (The canonical natural $ι (n) = n \cdot 1_{F}$ of a field, Canonical naturals are positive and strictly increasing, Inverses of positives are positive, and reciprocation reverses order, For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ).

[L3]

The alternating series test: a nonincreasing null sequence $(b_{k})$ makes $\sum ε_{k} b_{k}$ converge (The alternating series test: if $(b_{k})$ is nonincreasing with $b_{k} \to 0$ then $\sum_{k} (- 1)^{k} b_{k}$ converges, the sum lies between any two consecutive partial sums, and the error after $n$ terms is at most $b_{n}$ , Nondecreasing, increasing, nonincreasing, decreasing, monotone, and eventually monotone sequences, Limits and Cauchy sequences of reals, Series, partial sums, convergence and the sum, divergence, and the tail series).

[L4]

The $p$ -series theorem: $\sum_{k \geq 1} 1 / k^{p}$ converges if and only if $p > 1$ ; at $p = 1$ the harmonic series diverges (For rational $p > 0$ , $\sum 1 / k^{p}$ converges iff $p > 1$ , Rational powers $a^{r}$ of a positive base, Series, partial sums, convergence and the sum, divergence, and the tail series).

[L5]

Convergence in $R^{2}$ is componentwise (For $n \geq 1$ a sequence in $R^{n}$ converges iff each coordinate sequence converges, is Cauchy iff each coordinate sequence is Cauchy, and $R^{n}$ is complete in every norm clause 1, Series of vectors in $R^{n}$ , absolute convergence, rearrangement, and the set of rearrangement sums, The standard list $e : n \to F^{n}$ with $e_{i} (i) = 1_{F}$ and $e_{i} (j) = 0_{F}$ for $j \neq i$ is an ordered basis of $F^{n}$ ; hence $\dim_{F} F^{n} = n$ , and $F^{0}$ is the zero space with basis $\emptyset$ and dimension $0$ ).

[L6]

The inner product, the Euclidean norm and the definition of $Γ$ and $Γ^{⊥}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The $p$ -norms $∥ x ∥_{p}$ for rational $p \geq 1$ , and $∥ x ∥_{\infty}$ , A norm on a real vector space, the induced metric, and the dictionary with the metric axioms, The subspace $Γ$ of directions along which a series converges absolutely, and its orthogonal complement $Γ^{⊥}$ , Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , Integer powers $a^{m}$ ).

[L7]

For $c \neq 0$ , $\sum c a_{k}$ converges if and only if $\sum a_{k}$ converges (Convergent series add and scale termwise clause 3).

[L8]

The containment theorem (The set of rearrangement sums of a convergent series in $R^{n}$ is a nonempty subset of the affine subspace $s + Γ^{⊥}$ ) and the Riemann series theorem: a conditionally convergent real series has, for every real $c$ , a rearrangement converging to $c$ (The Riemann series theorem: a conditionally convergent real series has, for every $c \in R$ , a rearrangement with sum $c$ , and rearrangements diverging to $+ \infty$ , to $- \infty$ , and oscillating with any prescribed $lim inf \leq lim sup$ in $\overline{R}$ clause 1, Absolutely convergent and conditionally convergent series, and the general starting index, Injection, surjection, bijection).

Verification

technique · direct

1.1

$(b_{k})$ is positive, nonincreasing and converges to $0$ : positivity and monotonicity from $0 < ι (k + 1) < ι (k + 2)$ , and convergence because for a rational $ε > 0$ an index $K$ with $1 / ι (K + 1) < ε$ gives $b_{k} \leq b_{K} < ε$ for $k \geq K$ .

1.2

The second coordinate sequence is constantly $0$ , so its series converges with sum $0$ .

2.1

By the alternating series test $\sum_{k} c_{k} = \sum_{k} ε_{k} b_{k}$ converges; write $S$ for its sum.

step 1.1L3

2.2

$∥ x_{k} ∥_{2} = \sqrt{c_{k}^{2} + 0} = ∣ c_{k} ∣ = ∣ ε_{k} ∣ b_{k} = b_{k}$ , and $\sum_{k} b_{k}$ is the harmonic series, which diverges; so $\sum x_{k}$ does not converge absolutely, which is clause 2.

step 1.1L1L4L6

3.1

By componentwise convergence, $\sum x_{k}$ converges with sum $s = (S, 0)$ , which is clause 1.

step 2.1step 1.2L5

3.2

For $a = (a_{0}, a_{1}) \in R^{2}$ : $⟨ a, x_{k} ⟩ = a_{0} c_{k}$ , so $∣ ⟨ a, x_{k} ⟩ ∣ = ∣ a_{0} ∣ b_{k}$ . If $a_{0} = 0$ every term is $0$ and the series converges; if $a_{0} \neq 0$ then $∣ a_{0} ∣ > 0$ and convergence of $\sum_{k} ∣ a_{0} ∣ b_{k}$ would give convergence of $\sum_{k} b_{k}$ , which is false.

step 2.2L1L6L7

3.3

Conversely let $t \in R$ . The real series $\sum_{k} c_{k}$ converges by step 2.1 and does not converge absolutely by step 2.2, so it converges conditionally, and the Riemann series theorem supplies a bijection $σ$ of $N$ with $\sum_{k} c_{σ (k)} = t$ . The rearranged vector series $\sum_{k} x_{σ (k)}$ has first coordinate series $\sum_{k} c_{σ (k)}$ and second coordinate series constantly $0$ , so by componentwise convergence it converges to $(t, 0)$ ; hence $(t, 0) \in S (x)$ .

step 2.1step 2.2L5L8

4.1

Hence $Γ = {a : a_{0} = 0} = {(0, t) : t \in R}$ , the set of scalar multiples of $e_{1}$ .

step 3.2L6

5.1

For $y = (y_{0}, y_{1})$ : $y \in Γ^{⊥}$ means $⟨ (0, t), y ⟩ = t y_{1} = 0$ for every real $t$ , which at $t = 1$ forces $y_{1} = 0$ , and conversely $y_{1} = 0$ makes every such product $0$ . So $Γ^{⊥} = {(t, 0) : t \in R}$ , the set of scalar multiples of $e_{0}$ , and clause 3 is proved.

step 4.1L6

6.1

By the containment theorem, $S (x) \subseteq s + Γ^{⊥} = {(S + t, 0) : t \in R} = {(w, 0) : w \in R}$ .

step 3.1step 5.1L8

7.1

Steps 6.1 and 3.3 give $S (x) = s + Γ^{⊥}$ , which is clause 4.

step 6.1step 3.3∎

Remarks

Why this example is degenerate, and why that is said out loud. Every term $x_{k}$ lies in the line $Γ^{⊥}$ , so the whole series lives there and its rearrangement theory is the theory of the real series $\sum c_{k}$ . The equality in clause 4 is therefore the published The Riemann series theorem: a conditionally convergent real series has, for every $c \in R$ , a rearrangement with sum $c$ , and rearrangements diverging to $+ \infty$ , to $- \infty$ , and oscillating with any prescribed $lim inf \leq lim sup$ in $\overline{R}$ wearing two coordinates, not a higher-dimensional phenomenon. Nothing here supports or contradicts any statement about a series whose terms span $R^{2}$ .
The divergence of the harmonic series may be had two ways. Step 2.2 uses For rational $p > 0$ , $\sum 1 / k^{p}$ converges iff $p > 1$ at $p = 1$ ; the Cauchy condensation test gives the same conclusion, and either citation would do.
What the example makes concrete. $Γ$ is computed from its definition, one direction at a time, and turns out to be the set of directions orthogonal to where the series actually moves: testing against $e_{1}$ sees only zeros, and testing against $e_{0}$ sees the alternating harmonic series, which is not absolutely summable. That is exactly the dichotomy The subspace $Γ$ of directions along which a series converges absolutely, and its orthogonal complement $Γ^{⊥}$ is built to record.

Depends on

Used by

FALSE: if a convergent series in ℝⁿ does not converge absolutely, then every point of ℝⁿ is the sum of some rearrangement of it False statement

Dependency tree · two levels

137 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on 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)

A convergent series in R2R2 with ΓΓ a line and Γ⊥Γ⊥ a line, computed from the definition