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Question 775: 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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P1000-0775 · q-20261015-advance-20261117-r02-ratio-test-series-0100

Original question and givens

Use mode=zero_terms and positive integer scale=26. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).

  • G001 — Use mode=zero_terms and positive integer scale=26. 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=zero_terms; scale=26

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

a_n=26 for odd n; a_n=0 for even n

After

infinitely many denominators a_n are0

Conditions: none

Even indices occur arbitrarily far out, so there is no eventually nonzero tail.

Step identifiers and dependencies

Rule: zero_denominators · Uses: G001, S001

S003 · implies

Before

successive quotient at an even index

After

division by0: ratio undefined

Conditions: none

An undefined ratio is not a zero ratio, and a nonexistent limit cannot be labelled0.

Step identifiers and dependencies

Rule: ratio_not_defined · 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

(zero_terms, 26, a_n=26_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, 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: b0c883f85c9c701c0d0d846072c25e03f2dee44f727d8f25034e29617d4f896d
  • candidate record SHA-256: fe395a3b61a5448bf5dad97772fea4de2538b3d650018c1dd8a0801139d609ab
  • semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772
  • question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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