Absolute and Conditional Convergence; Rearrangement; Products

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1 · Prerequisites

Real Analysis

Set Theory

2 · Logical flowchart

3 · Items

Definition

Absolutely convergent and conditionally convergent series, and the general starting index

Lemma

Positive and negative parts: aₖ = aₖ⁺ - aₖ⁻ and |aₖ| = aₖ⁺ + aₖ⁻; a series converges absolutely iff both ∑ aₖ⁺ and ∑ aₖ⁻ converge, and for a conditionally convergent series both diverge to +∞

Lemma

Abel summation by parts: with Aₙ = ∑_k<n aₖ one has ∑_k<n aₖ bₖ = Aₙ bₙ₋₁ - ∑_k < n-1 Aₖ₊₁ (bₖ₊₁ - bₖ) for every n ≥ 1

Theorem

Dirichlet's test: if the partial sums of ∑ aₖ are bounded and (bₖ) is nonincreasing with bₖ → 0, then ∑ aₖ bₖ converges

Theorem

The alternating series test: if (bₖ) is nonincreasing with bₖ → 0 then ∑ₖ (-1)ᵏ bₖ converges, the sum lies between any two consecutive partial sums, and the error after n terms is at most bₙ

Theorem

Abel's test: if ∑ aₖ converges and (bₖ) is monotone and bounded then ∑ aₖ bₖ converges

Definition

Rearrangement of a series along a bijection of ℕ, and unconditional convergence

Theorem

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

Theorem

The Riemann series theorem: a conditionally convergent real series has, for every c ∈ ℝ, a rearrangement with sum c, and rearrangements diverging to +∞, to -∞, and oscillating with any prescribed liminf ≤ limsup in ℝ̄

Corollary

For a series of real numbers, unconditional convergence and absolute convergence are the same property

Theorem

Grouping: if ∑ aₖ converges and (nⱼ) is strictly increasing with n₀ = 0, the series of blocks ∑_k=nⱼ^nⱼ₊₁-1 aₖ converges to the same sum

Definition

Fubini for double series: if ∑ᵢ ∑ⱼ |aᵢⱼ| converges then both iterated sums and the sum along every bijection ℕ → ℕ × ℕ converge to one and the same value

Definition

Infinite products: partial products, and convergence to a nonzero limit after finitely many vanishing factors

Theorem

For pₖ ≥ 0 the product ∏ (1 + pₖ) converges iff ∑ pₖ converges, with 1 + ∑_k<n pₖ ≤ ∏_k<n(1+pₖ) ≤ 1/(1 - ∑_k<n pₖ) when ∑_k<n pₖ < 1; for 0 ≤ pₖ < 1 the product ∏ (1 - pₖ) converges iff ∑ pₖ converges and its partial products tend to 0 otherwise; and ∑ |pₖ| convergent implies ∏ (1+pₖ) convergent

Theorem

Base-b expansions: for an integer b ≥ 2 every x ∈ [0,1) is the sum of ∑_j ≥ 0 dⱼ / b^ j+1 for digits dⱼ < b, and the digit sequence is unique among those that are not eventually constantly b-1

Remark

The same question in ℝᵈ: what the set of rearrangement sums looks like, and why that answer is not reachable at this point in the reading order

Remark

Selected sums and products on this page that are proved to exist without being evaluated, and what their evaluation waits for

False statement

Absolute and Conditional Convergence; Rearrangement; Products

3 · Items

Absolutely convergent and conditionally convergent series, and the general starting index

Positive and negative parts: aₖ = aₖ⁺ - aₖ⁻ and |aₖ| = aₖ⁺ + aₖ⁻; a series converges absolutely iff both ∑ aₖ⁺ and ∑ aₖ⁻ converge, and for a conditionally convergent series both diverge to +∞

Abel summation by parts: with Aₙ = ∑_k<n aₖ one has ∑_k<n aₖ bₖ = Aₙ bₙ₋₁ - ∑_k < n-1 Aₖ₊₁ (bₖ₊₁ - bₖ) for every n ≥ 1

Dirichlet's test: if the partial sums of ∑ aₖ are bounded and (bₖ) is nonincreasing with bₖ → 0, then ∑ aₖ bₖ converges

The alternating series test: if (bₖ) is nonincreasing with bₖ → 0 then ∑ₖ (-1)ᵏ bₖ converges, the sum lies between any two consecutive partial sums, and the error after n terms is at most bₙ

Abel's test: if ∑ aₖ converges and (bₖ) is monotone and bounded then ∑ aₖ bₖ converges

Rearrangement of a series along a bijection of ℕ, and unconditional convergence

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

The Riemann series theorem: a conditionally convergent real series has, for every c ∈ ℝ, a rearrangement with sum c, and rearrangements diverging to +∞, to -∞, and oscillating with any prescribed liminf ≤ limsup in ℝ̄

For a series of real numbers, unconditional convergence and absolute convergence are the same property

Grouping: if ∑ aₖ converges and (nⱼ) is strictly increasing with n₀ = 0, the series of blocks ∑_k=nⱼ^nⱼ₊₁-1 aₖ converges to the same sum

The Cauchy product of two series: cₙ = ∑ₖ₌₀ⁿ aₖ bₙ₋ₖ

Mertens' theorem: if ∑ aₖ converges absolutely to A and ∑ bₖ converges to B, their Cauchy product converges to AB

If ∑ aₖ and ∑ bₖ both converge absolutely then their Cauchy product converges absolutely, with sum AB

Fubini for double series: if ∑ᵢ ∑ⱼ |aᵢⱼ| converges then both iterated sums and the sum along every bijection ℕ → ℕ × ℕ converge to one and the same value

Infinite products: partial products, and convergence to a nonzero limit after finitely many vanishing factors

Base-b expansions: for an integer b ≥ 2 every x ∈ [0,1) is the sum of ∑_j ≥ 0 dⱼ / b^ j+1 for digits dⱼ < b, and the digit sequence is unique among those that are not eventually constantly b-1

The same question in ℝᵈ: what the set of rearrangement sums looks like, and why that answer is not reachable at this point in the reading order

Selected sums and products on this page that are proved to exist without being evaluated, and what their evaluation waits for

FALSE: every convergent series converges absolutely

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

FALSE: the Cauchy product of two convergent series converges

FALSE: if some grouping of a series converges then the series itself converges

FALSE: whenever both iterated sums of a double array exist, they are equal

FALSE: ∏ (1 + pₖ) converges whenever pₖ → 0