Alphabeta Math

Series: Convergence and the Nonnegative Tests: Examples and Counterexamples

About this page
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 12 results · all verified · 12 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs; all 12 also cleared it.
1 · Prerequisites
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

The harmonic series ∑ 1/k diverges, by condensation and by Oresme block grouping

Example

∑ 1/k² converges with sum at most 2, by comparison with the telescoping ∑ 1/(k(k-1))

Example

Geometric sums computed: ∑_k ≥ 1 2⁻ᵏ = 1 and ∑_k ≥ 0 (-1/3)ᵏ = 3/4

Example

∑_k ≥ 1 1/(k(k+1)) = 1

Example

Condensation reduces ∑ 1/kᵖ to a geometric series with ratio 2¹⁻ᵖ

Example

Abel-Dini applied to ∑ 1/k: ∑ 1/(k sₖ) still diverges while ∑ 1/(k sₖ²) converges

Example

A series with ratio limit exactly 1 that Raabe decides

Counterexample

aₖ = 2^-k+(-1)ᵏ has ratio limsup 2 and liminf 1/8, so the ratio test fails, while the root test gives convergence

Counterexample

∑ k^-1/2 diverges and ∑ k⁻² converges, and both have root limit exactly 1

Counterexample

Two series with aₖ ≤ bₖ for all k, ∑ bₖ convergent and ∑ aₖ divergent, when the terms may be negative

Counterexample

A nonnegative non-monotone sequence for which ∑ aₖ and ∑ 2ᵏ a_2ᵏ behave differently

Counterexample

With aₖ/bₖ → 0, convergence of ∑ aₖ does not give convergence of ∑ bₖ

Sources