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Question 760: Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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P1000-0760 · q-20261015-advance-20261117-r02-ratio-test-series-0085

Original question and givens

Use mode=finite_samples and positive integer scale=11. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).

  • G001 — Use mode=finite_samples and positive integer scale=11. 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.

Step-by-step derivation

S001 · implies

Before

boundary mode=finite_samples; scale=11

After

check the precise hypothesis that fails

Conditions: none

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

Step identifiers and dependencies

Rule: ratio_hypothesis_audit · Uses: G001

S002 · implies

Before

ratios=1/2 only for n≤11

After

declared later ratios=2

Conditions: none

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

Step identifiers and dependencies

Rule: favorable_prefix · Uses: G001, S001

S003 · implies

Before

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 dependencies

Rule: prefix_counterexample_tail · Uses: G001, S002

S004 · implies

Before

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 dependencies

Rule: 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, 11, ratios_equal_1/2_for_n<=11_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: aec6ed5f75119d385b5b3d70ede440b753ca64170dd608fa582891b8e6268c69
  • candidate record SHA-256: 557deec88fa1a3f70c258d5ccc100db3f6fb3ff8e0df186886f9de97daa29efe
  • semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772
  • question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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