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R0052-26985 · q-20261015-advance-20261117-r01-series-comparison-test-0010
Use exact scales A=10,B=11, rational ratio r=5/11. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).
G001 — Use exact scales A=10,B=11, rational ratio r=5/11. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
source roles = [series-upper-geometric]
After
derived = [[a_scale, b_scale, ratio, sum_a, sum_b, sums_ordered, certificate]]
Conditions: none
List this operation/mode exact source field order without JavaScript.Keep Boolean,string and null roles distinct;do not pad fields or infer the interface from answer shape.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [10, 11, 5/11]
After
derived = [EVERY tail term is A*r^k and B*r^k for k>=0, r^0=1 including r=0, Yes]
Conditions: none
The whole tail law is given,with k starting at0.Nonnegative A<=B and r give termwise comparison;0<=r<1 licenses the geometric sum.At r=0 the first term still uses r^0=1.
Rule: Series_WHOLE_geometric_tail_premises_not_finite_samples · Uses: G001, S001
Before
Series ORIGINAL one minus ratio := sub(1, 5/11)
After
Series ORIGINAL one minus ratio = 6/11
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Series ORIGINAL geometric tailA := div(10, 6/11)
After
Series ORIGINAL geometric tailA = 55/3
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
Series ORIGINAL geometric tailB := div(11, 6/11)
After
Series ORIGINAL geometric tailB = 121/6
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
source roles = [10, 11, 5/11, 6/11]
After
derived = [55/3, 121/6, Yes]
Conditions: none
For the finite sum,multiplication by(1-r) telescopes to A(1-r^N);r^N tends to0,hence the tail sum A/(1-r),and similarly for B.The nonnegative a partial sums increase and are bounded by the convergent b sum.This is not a finite-sample replacement for all n.
Rule: Series_GENERAL_geometric_partial_sum_limit_and_upper_comparison · Uses: G001, S005
S006 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.S005 · premise_and_result_audit — Recompute Series ORIGINAL geometric tailB with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute Series ORIGINAL geometric tailA with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute Series ORIGINAL one minus ratio with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.S001 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.(10, 11, 5/11, 55/3, 121/6, Yes, upper_comparison_proves_convergence)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 7d4daa200fed225f396f9846ff33760622490d65e2f88cf1abe816cac8d7076acandidate record SHA-256: da99791427153d4aa64096c89a54365e5f8959ae7d5f6134a4ada6b9d95e2a88semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2Lean: NOT_RUN · human review: NOT_PERFORMED