Comparison tests for series

Mathematical statement

For every sufficiently large n require0≤a_n≤b_n. Convergence of the b series implies convergence of the a series; divergence of the a series implies divergence of the b series. Exact families use geometric tails, arbitrary finite prefixes and positive constant lower bounds. Finitely many terms do not change convergence, but every tail comparison is a universal obligation. Negative terms, a reversed inequality, finitely sampled comparisons or a divergent upper bound do not supply the required test.

Representative formula (exact TeX)

C(a,b)=c

Assumptions

• A declared cutoff makes the nonnegative comparison true for every later index.
• The reference series has its own convergence or divergence certificate, not a guess from a finite prefix.

Symbol scope

• C — series-comparison operator
• a — compared series terms
• b — reference series terms
• c — comparison conclusion

Intuition, definition, and misconceptions

Separate convergent upper comparison from divergent lower comparison.

Eventual comparison allows arbitrary finite prefixes but requires the entire tail.

Geometric tails and positive lower bounds are reproducible instances, not the general theorem proof.

Finite samples, reversed inequalities or a divergent upper bound are not comparison-test certificates.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Infinite series — Use series and partial-sum semantics.
• Usual order on the real numbers — Keep the termwise inequality direction.
• Sequence limits and exact tail cutoffs — Retain the partial-sum convergence background.
• Monotone bounded sequence convergence theorem — Use monotone nonnegative partial sums and bounds.

Verification boundary

Exact geometric instances certify their specified sequences; general comparison still needs the full quantified argument.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add limit comparison and the integral test.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.