Question 770: Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises

Original question and givens

Use mode=signed_without_absolute and positive integer scale=21. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).

• G001 — Use mode=signed_without_absolute and positive integer scale=21. 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

Step-by-step derivation

boundary mode=signed_without_absolute; scale=21

check the precise hypothesis that fails

a_n=(-1)^n·21·(1/2)^n

a_(n+1)/a_n=−1/2

|a_(n+1)/a_n|

absolute ratio=1/2<1

missing or misused premise

this attempted test is not licensed

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

(signed_without_absolute, 21, a_n=(-1)^n*21*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, 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.