# Ratio test for series

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

For eventually nonzero terms, an absolute successive-ratio limit below1 gives absolute convergence, above1 gives divergence, and equal to1 is inconclusive.

mko-ratio-test-series · theorem · v0.10.0



## Mathematical statement

Declare the existing limit L=lim|a_(n+1)/a_n| on an eventually nonzero tail. When L<1 choose L<q<1 and establish eventual geometric domination to obtain absolute convergence. When L>1 the term magnitudes eventually fail to tend to0, so the series diverges. L=1 gives no decision: reciprocal-square and harmonic tails have opposite statuses with that same limit. Use absolute ratios and an actual limit argument, not a finite favorable prefix. Zero denominators and missing limits are separate boundary cases.





## Representative formula (exact TeX)

R(a,L)=c





## Assumptions


- Successive ratios are defined on some whole tail and their absolute-value limit exists.
- Comparison of L with1 uses exact rational values or a declared symbolic limit certificate.





## Symbol scope


- R — ratio-test operator
- a — series terms
- L — absolute-ratio limit
- c — ratio-test conclusion





## Intuition, definition, and misconceptions

Qualify eventual nonzero terms and the absolute-ratio limit first.

For L<1 retain L<q<1 and domination; separate L>1 from L=1.

Nonconstant reciprocal-power ratios give opposite L=1 witnesses, while finite prefixes and nonexistent limits do not qualify.

L=1 does not imply convergence, and a few signed ratios do not establish the needed absolute limit.





## Adopted derivation guide

[Ratio limits, geometric domination and L=1](https://ocme.evemisslab.com/en/derivations/derivation-mko-ratio-test-series-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Absolute convergence of series](https://ocme.evemisslab.com/en/mko/mko-absolute-convergence-of-series/) — Use the L<1 conclusion and series semantics.
- [Sequence limits and exact tail cutoffs](https://ocme.evemisslab.com/en/mko/mko-sequence-limit/) — Require the ratio limit.
- [Absolute value](https://ocme.evemisslab.com/en/mko/mko-absolute-value/) — Use absolute rather than signed ratios.
- [Comparison tests for series](https://ocme.evemisslab.com/en/mko/mko-series-comparison-test/) — Obtain geometric domination after choosing q.





## Verification boundary

Geometric and reciprocal-power instances do not replace the general ratio-test proof. The current additive source revision writes the entire rational base in parentheses in A*(r)^n/(n+shift); original ambiguous displays remain in the retained historical source, not the active answers.

not_formalized

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





## Next obligation

Add the root test and limsup versions.





## 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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/ratio-test-series.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-ratio-test-series/index.md)