A convergent series in $\mathbb{R}^{2}$ with $\Gamma$ a line and $\Gamma^{\perp}$ a line, computed from the definition

Let $(\varepsilon_k)$ be the alternating sequence, the unique sequence of reals with $\varepsilon_0 = 1$ and $\varepsilon_{k+1} = -\varepsilon_k$, so $|\varepsilon_k| = 1$ for every $k$ (The even and odd index maps and the alternating sequence: strictly increasing $e, o$ with $\mathbb{N}$ their disjoint union, and the unique $(s_k)$ with $s_0 = 1$, $s_{\sigma(k)} = -s_k$, which satisfies $|s_k| = 1$, $s \circ e \equiv 1$ and $s \circ o \equiv -1$). In $\mathbb{R}^{2}$ put

$x_k \;:=\; \Bigl(\frac{\varepsilon_k}{\iota(k+1)},\ 0\Bigr) \qquad (k \in \mathbb{N}),$

with $\iota$ the canonical natural (The canonical natural $\iota(n) = n \cdot 1_F$ 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:

1. $\sum x_k$ 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.
2. $\sum x_k$ does not converge absolutely (Series of vectors in $\mathbb{R}^n$, absolute convergence, rearrangement, and the set of rearrangement sums).
3. $\Gamma = \{\, (0,t) : t \in \mathbb{R} \,\}$, the line of multiples of $e_1$, and $\Gamma^{\perp} = \{\, (t,0) : t \in \mathbb{R} \,\}$, the line of multiples of $e_0$ (The subspace $\Gamma$ of directions along which a series converges absolutely, and its orthogonal complement $\Gamma^{\perp}$).
4. Consequently The set of rearrangement sums of a convergent series in $\mathbb{R}^n$ is a nonempty subset of the affine subspace $s + \Gamma^{\perp}$ confines every rearrangement sum to the horizontal line $s + \Gamma^{\perp} = \{(t,0) : t \in \mathbb{R}\}$; and for this series the confinement is exact, $\mathcal{S}(x) = s + \Gamma^{\perp}$, by 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 $\overline{\mathbb{R}}$ 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 $\mathcal{S}(x) = s + \Gamma^{\perp}$ for a series genuinely spread over $\mathbb{R}^{n}$ with $n \ge 2$, a question this library does not settle (Conventions of this page, the standing $n \ge 1$ hypothesis, and what is taken up elsewhere in the reading order).

• Why this example is degenerate, and why that is said out loud. Every term $x_k$ lies in the line $\Gamma^{\perp}$, 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 $c \in \mathbb{R}$, a rearrangement with sum $c$, and rearrangements diverging to $+\infty$, to $-\infty$, and oscillating with any prescribed $\liminf \le \limsup$ in $\overline{\mathbb{R}}$ wearing two coordinates, not a higher-dimensional phenomenon. Nothing here supports or contradicts any statement about a series whose terms span $\mathbb{R}^{2}$.

• 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. $\Gamma$ 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 $\Gamma$ of directions along which a series converges absolutely, and its orthogonal complement $\Gamma^{\perp}$ is built to record.