False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-27

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FALSE: every rearrangement of a convergent series converges, and to the same sum

Statement

False claim: for every sequence $(a_k)$ of reals whose series converges (Series, partial sums, convergence and the sum, divergence, and the tail series) and every bijection $\sigma : \mathbb{N} \to \mathbb{N}$ , the rearranged series $\sum a_{\sigma(k)}$ (Rearrangement of a series along a bijection of $\mathbb{N}$ , and unconditional convergence) converges, with the same sum.

What is true is that hypothesis: the claim holds for absolutely convergent series, and that is Dirichlet's rearrangement theorem: an absolutely convergent series converges unconditionally, and every rearrangement of it has the same sum. Dropping "absolutely" makes it false in both of its assertions at once, and the same witness refutes both.

Let $(\varepsilon_j)$ be the alternating sequence (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 $s \circ o \equiv -1$ ) and put $a_j := \varepsilon_j/\iota(j+1)$ , the alternating harmonic series. It converges, by the alternating series test, and does not converge absolutely, its series of absolute values being the harmonic series (For rational $p > 0$ , $\sum 1/k^p$ converges iff $p > 1$ ). So it converges conditionally, and 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 $\overline{\mathbb{R}}$ applies to it.

Facts & Assumptions

Given: The alternating sequence $(\varepsilon_j)$ , the sequence $b_j := 1/\iota(j+1)$ , and $a_j := \varepsilon_j b_j$ , whose series is the alternating harmonic series.

[A1]

The refuted claim: for every convergent series of reals and every bijection of $\mathbb{N}$ , the rearranged series converges with the same sum.

[L1]

$|\varepsilon_j| = 1$ for every $j$ (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 $s \circ o \equiv -1$ ).

[L2]

The canonical naturals $\iota(n)$ are positive for $n \ge 1$ and strictly increasing; if $0 < u < v$ then $0 < 1/v < 1/u$ ; and for every real $\varepsilon > 0$ there is $n \ge 1$ with $1/\iota(n) < \varepsilon$ (Canonical naturals are positive and strictly increasing, Inverses of positives are positive, and reciprocation reverses order, For every $\varepsilon > 0$ in a complete ordered field there is a natural $n \ge 1$ with $1/n < \varepsilon$ ).

[L3]

The alternating series test (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).

[L4]

$\sum_{k\ge1} 1/k^{p}$ converges if and only if $p > 1$ , with $\iota(k)^{1} = \iota(k)$ ; and $\sum_{k \ge 1} x_k$ is the series of $j \mapsto x_{j+1}$ (For rational $p > 0$ , $\sum 1/k^p$ converges iff $p > 1$ , Rational powers $a^r$ of a positive base, Existence and uniqueness of $n$ -th roots: a unique $a^{1/n} \ge 0$ with $(a^{1/n})^n = a$ , Integer powers $a^m$ , Series, partial sums, convergence and the sum, divergence, and the tail series).

[L5]

Absolute value: $|xy| = |x|\,|y|$ (Basic properties of the absolute value).

[L6]

Absolute and conditional convergence (Absolutely convergent and conditionally convergent series, and the general starting index).

[L7]

The Riemann series theorem: a conditionally convergent series has, for every real $c$ , a rearrangement converging to $c$ , and one whose partial sums diverge to $+\infty$ (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 $\overline{\mathbb{R}}$ , Divergence to $+\infty$ and to $-\infty$ ).

[L8]

An absolutely convergent series converges unconditionally (Dirichlet's rearrangement theorem: an absolutely convergent series converges unconditionally, and every rearrangement of it has the same sum).

Refutation

technique · direct

1.1

The sequence $(b_j)$ is positive, nonincreasing and converges to $0$ : positivity and monotonicity from $0 < \iota(j+1) < \iota(j+2)$ , and convergence because, given a rational $\varepsilon > 0$ , an $n \ge 1$ with $1/\iota(n) < \varepsilon$ satisfies $b_j \le 1/\iota(n) < \varepsilon$ for every $j \ge n$ .

givenL2

2.1

By the alternating series test $\sum a_j$ converges; write $S$ for its sum.

step 1.1L3

2.2

For every $j$ , $|a_j| = |\varepsilon_j| b_j = 1/\iota(j+1)$ , and $\sum_j 1/\iota(j+1)$ is the $p$ -series $\sum_{k\ge1}1/k$ at $p = 1$ , which diverges.

step 1.1L1L4L5

3.1

So $\sum a_j$ converges conditionally.

step 2.1step 2.2L6

4.1

By the Riemann series theorem there is a bijection $\sigma$ of $\mathbb{N}$ with $\sum a_{\sigma(k)}$ convergent of sum $S + 1$ , a number different from $S$ .

step 3.1L7

4.2

By the same theorem there is a bijection $\tau$ of $\mathbb{N}$ for which the partial sums of $\sum a_{\tau(k)}$ diverge to $+\infty$ , so that rearranged series does not converge at all.

step 3.1L7

5.1

The claim [A1] therefore fails twice over for the alternating harmonic series: once in its assertion that the sum is preserved, by step 4.1, and once in its assertion that the rearranged series converges, by step 4.2.

step 4.1step 4.2A1

6.1

The claim is false. What is true is the same statement with "converges" strengthened to "converges absolutely" in the hypothesis.

step 5.1A1L8∎

Remarks

Neither half of the claim survives. It is often stated as though the only risk were a change of value; step 4.2 shows the rearranged series may fail to converge, and 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 $\overline{\mathbb{R}}$ shows the partial sums may be made to oscillate between any two prescribed extended reals.
The hypothesis that repairs the claim is exactly the right one. By For a series of real numbers, unconditional convergence and absolute convergence are the same property, absolute convergence is not merely sufficient for the conclusion but necessary: a convergent series all of whose rearrangements converge is absolutely convergent. So there is no intermediate hypothesis to look for.
What is fixed and what is not. The terms of the series are fixed; only the order changes. That an infinite sum should depend on the order at all is the point of the example, and it is why Series, partial sums, convergence and the sum, divergence, and the tail series defines the sum as the limit of the partial sums of a sequence, not as a sum over a set of indices.

Depends on

Used by

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Sources

Riemann series theorem (Wikipedia) (standard reference, not scraped)
Harmonic series (mathematics) (Wikipedia) (standard reference, not scraped)
N. Donaldson, Math 140A: Real Analysis notes (standard reference, not scraped)
John K. Hunter, An Introduction to Real Analysis, Chapter 4 (standard reference, not scraped)