Alphabeta Math

Series: Convergence and the Nonnegative Tests

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✓ 23 results · all verified · 18 also independently AI-judged
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1 · Prerequisites
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Definition

Series, partial sums, convergence and the sum, divergence, and the tail series

Lemma

A series converges iff each of its tail series converges, and the sum splits as s_N plus the N-th tail

Lemma

If a series converges then its terms tend to 0

Theorem

A series converges iff for every ε > 0 there is N with |aₘ₊₁ + … + aₙ| < ε for all n > m ≥ N

Lemma

Convergent series add and scale termwise

Lemma

If ∑ |aₖ| converges then ∑ aₖ converges

Theorem

For |r| < 1, ∑_k ≥ 0 rᵏ = 1/(1-r), and for |r| ≥ 1 the series diverges

Lemma

∑ (bₖ - bₖ₊₁) converges iff (bₖ) converges, with sum b₀ - lim bₖ

Theorem

A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum

Theorem

If 0 ≤ aₖ ≤ bₖ eventually, convergence of ∑ bₖ gives convergence of ∑ aₖ, and divergence of ∑ aₖ gives divergence of ∑ bₖ

Theorem

For aₖ, bₖ > 0 with aₖ/bₖ → L: if L ∈ (0,∞) the two series share their behaviour, while L = 0 and L = ∞ give one implication each

Theorem

For a nonincreasing nonnegative sequence, ∑ aₖ converges iff ∑ 2ᵏ a_2ᵏ converges

Theorem

For rational p > 0, ∑ 1/kᵖ converges iff p > 1

Theorem

Root test: limsup |aₖ|^1/k < 1 gives absolute convergence and hence convergence, > 1 gives divergence, and = 1 decides nothing

Theorem

Ratio test: limsup |aₖ₊₁/aₖ| < 1 gives absolute convergence and hence convergence, and liminf |aₖ₊₁/aₖ| > 1 gives divergence

Corollary

Whenever the ratio test decides, the root test decides the same way, and the converse fails

Theorem

Kummer: for positive terms aₖ and weights ζₖ > 0, liminf(ζₖ aₖ/aₖ₊₁ - ζₖ₊₁) > 0 gives convergence, and if ∑ 1/ζₖ diverges while that expression is eventually ≤ 0 the series diverges

Corollary

Kummer with ζₖ = 1 recovers the ratio test

Corollary

Raabe is Kummer with ζₖ = k+1: for positive terms, liminf (k+1)(aₖ/aₖ₊₁ - 1) > 1 gives convergence and limsup < 1 gives divergence

Theorem

Gauss: for positive terms, if aₖ/aₖ₊₁ = 1 + h/k + rₖ with |rₖ| ≤ C k^-1-ε for k ≥ 1, some constant C and some rational ε > 0, the series converges iff h > 1

Theorem

For a divergent series of positive terms with partial sums sₖ, the series ∑ aₖ/sₖ diverges and ∑ aₖ/sₖ² converges

Remark

How the nonnegative tests are ordered by strength, and which of them this page cannot state without the logarithm

False statement

FALSE: if aₖ → 0 then ∑ aₖ converges

False statement

FALSE: limsup |aₖ₊₁/aₖ| ≥ 1 implies the series diverges

False statement

FALSE: there is a divergent series of positive terms that diverges more slowly than every other, hence a universal comparison test

Sources