# Question 764: Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

P1000-0764 · q-20261015-advance-20261117-r02-ratio-test-series-0089



## Original question and givens

Use mode=finite_samples and positive integer scale=15. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).


- G001 — Use mode=finite_samples and positive integer scale=15. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).





## Definition and scope

The ratio test needs an eventually nonzero tail, absolute ratios and an existing limit. Boundary items name the failed hypothesis without deciding the series.


- [mko-ratio-test-series](https://ocme.evemisslab.com/en/mko/mko-ratio-test-series/)





## Step-by-step derivation

S001 · impliesBefore

boundary mode=finite_samples; scale=15

After

check the precise hypothesis that fails

Conditions: none

Identify the failing hypothesis instead of declaring every boundary case divergent.

Step identifiers and dependenciesRule: ratio_hypothesis_audit · Uses: G001





S002 · impliesBefore

ratios=1/2 only for n≤15

After

declared later ratios=2

Conditions: none

Later ratios can change; the question supplies a counterexample tail that does change.

Step identifiers and dependenciesRule: favorable_prefix · Uses: G001, S001





S003 · impliesBefore

after the finite prefix, multiply magnitude by2 each step

After

terms eventually grow instead of tending to0

Conditions: none

A positive initial term with the declared recurrence has a favorable prefix followed by a doubling tail.

Step identifiers and dependenciesRule: prefix_counterexample_tail · Uses: G001, S002





S004 · impliesBefore

missing or misused premise

After

this attempted test is not licensed

Conditions: none

Retain the original boundary verdict without adding an unsupported general convergence conclusion.

Step identifiers and dependenciesRule: boundary_verdict · Uses: G001, S003









## Reverse checks


- S004 · premise_and_result_audit — Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values.
- S003 · premise_and_result_audit — Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values.
- S002 · premise_and_result_audit — Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values.
- S001 · premise_and_result_audit — Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values.





## Result

(finite_samples, 15, ratios_equal_1/2_for_n<=15_then_equal_2, finite_prefix_cannot_determine_ratio_limit, No)

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: aabd88f93d44c6a0d4f47af3da1f72d8a948aec1f504a13da95d3f42d0988318
- candidate record SHA-256: 885e9b2ff2b79e76508c6639e4a941d767c4daab870e1f201aa59b3650e278b9
- semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772
- question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b
- Lean: NOT_RUN · human review: NOT_PERFORMED

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