ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-27

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Every rearrangement of $\sum_{k \geq 0} (- 1 / 2)^{k}$ converges to $2 / 3$

Example

Let $r : = - 1 / 2$ and consider $\sum_{k \geq 0} r^{k}$ , with $r^{k}$ the integer power (Integer powers $a^{m}$ ), so that the first term is $r^{0} = 1$ . Then:

$\sum_{k = 0}^{\infty} (- \frac{1}{2})^{k} = \frac{1}{1 - (- 1 / 2)} = \frac{2}{3},$

the series converges absolutely (Absolutely convergent and conditionally convergent series, and the general starting index), and every rearrangement of it along a bijection of $N$ (Rearrangement of a series along a bijection of $N$ , and unconditional convergence) converges, again to $2 / 3$ .

This is the contrast case for the whole page. The alternating harmonic series ( $\sum_{j \geq 0} (- 1)^{j} / (j + 1)$ converges conditionally, with sum strictly between $1 / 2$ and $1$ ) has terms with the same alternating sign pattern, tending to $0$ just as these do, and can be rearranged to any real whatever; this series cannot be rearranged to anything but $2 / 3$ . The difference is absolute convergence and nothing else, by For a series of real numbers, unconditional convergence and absolute convergence are the same property.

Facts & Assumptions

Given: $r = - 1 / 2$ and the sequence $a_{k} : = r^{k}$ (Integer powers $a^{m}$ ).

[L1]

Geometric series: for $∣ x ∣ < 1$ the series $\sum x^{k}$ converges with sum $1 / (1 - x)$ , the series starting at $k = 0$ with first term $x^{0} = 1$ (For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges, Series, partial sums, convergence and the sum, divergence, and the tail series).

[L2]

Absolute value: $∣ x y ∣ = ∣ x ∣ ∣ y ∣$ , $∣ 1 ∣ = 1$ , and $∣ - 1 / 2 ∣ = 1 / 2$ (Basic properties of the absolute value).

[L3]

Powers: $x^{0} = 1$ and $x^{n + 1} = x^{n} x$ (Integer powers $a^{m}$ , Laws of integer exponents).

[L4]

The principle of induction on $N$ (The principle of mathematical induction).

[L5]

An absolutely convergent series converges unconditionally: every rearrangement converges to the same sum (Dirichlet's rearrangement theorem: an absolutely convergent series converges unconditionally, and every rearrangement of it has the same sum, Rearrangement of a series along a bijection of $N$ , and unconditional convergence, Absolutely convergent and conditionally convergent series, and the general starting index).

Verification

technique · direct

1.1

An induction gives $∣ r^{k} ∣ = ∣ r ∣^{k} = (1 / 2)^{k}$ for every $k$ : at $k = 0$ both sides are $1$ , and $∣ r^{k + 1} ∣ = ∣ r^{k} r ∣ = ∣ r^{k} ∣ ∣ r ∣ = (1 / 2)^{k} (1 / 2)$ .

L2L3L4

1.2

Since $∣ r ∣ = 1 / 2 < 1$ , the series $\sum_{k} r^{k}$ converges with sum $1 / (1 - r) = 1 / (3 / 2) = 2 / 3$ .

L1L2algebra

2.1

Since $∣ 1 / 2 ∣ = 1 / 2 < 1$ , the series $\sum_{k} ∣ r^{k} ∣ = \sum_{k} (1 / 2)^{k}$ converges, with sum $1 / (1 - 1 / 2) = 2$ ; so $\sum_{k} r^{k}$ converges absolutely.

step 1.1L1L2

3.1

By Dirichlet's rearrangement theorem, for every bijection $σ$ of $N$ the series $\sum_{k} r^{σ (k)}$ converges, with the same sum $2 / 3$ .

step 2.1L5

4.1

So the series converges absolutely with sum $2 / 3$ , and every rearrangement of it converges to $2 / 3$ .

step 1.2step 2.1step 3.1∎

Remarks

The starting index matters and is stated. The series begins at $k = 0$ , and its first term is $r^{0} = 1$ ; the same series started at $k = 1$ would sum to $2 / 3 - 1 = - 1 / 3$ . For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges makes the same point for $\sum 2^{- k}$ , whose sum is $2$ from index $0$ and $1$ from index $1$ .
Nothing is checked bijection by bijection. The point of Dirichlet's rearrangement theorem: an absolutely convergent series converges unconditionally, and every rearrangement of it has the same sum is that no property of $σ$ beyond bijectivity is used; the sum of the absolute values, here $2$ , is what bounds every partial sum of every rearrangement.
The signs are a red herring. The same conclusion holds for $\sum (1 / 2)^{k}$ , whose terms are all positive; a series of nonnegative terms that converges is absolutely convergent, so no such series has an interesting rearrangement theory. The alternating signs here are chosen only to make the comparison with the alternating harmonic series exact.

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Sources

Geometric series (Wikipedia) (standard reference, not scraped)
Absolute convergence (Wikipedia) (standard reference, not scraped)
John K. Hunter, An Introduction to Real Analysis (standard reference, not scraped)
N. Donaldson, Math 140A: Series (standard reference, not scraped)

Every rearrangement of ∑k≥0(−1/2)k∑k≥0​(−1/2)k converges to 2/32/3