# Question 26981: A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series [Return to catalogue page 135](https://ocme.evemisslab.com/en/questions/page/135/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261117-r01-series-comparison-test-foundations/#q-20261015-advance-20261117-r01-series-comparison-test-0006) 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. R0052-26981 · q-20261015-advance-20261117-r01-series-comparison-test-0006 ## Original question and givens Use exact scales A=6,B=7, rational ratio r=1/7. 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=6,B=7, rational ratio r=1/7. 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). ## Definition and scope 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. - [mko-series-comparison-test](https://ocme.evemisslab.com/en/mko/mko-series-comparison-test/) ## Step-by-step derivation S001 · impliesBefore 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. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [6, 7, 1/7] 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. Step identifiers and dependenciesRule: Series_WHOLE_geometric_tail_premises_not_finite_samples · Uses: G001, S001 S003 · calculationBefore Series ORIGINAL one minus ratio := sub(1, 1/7) After Series ORIGINAL one minus ratio = 6/7 Conditions: none Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Series ORIGINAL geometric tailA := div(6, 6/7) After Series ORIGINAL geometric tailA = 7 Conditions: none Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Series ORIGINAL geometric tailB := div(7, 6/7) After Series ORIGINAL geometric tailB = 49/6 Conditions: none Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [6, 7, 1/7, 6/7] After derived = [7, 49/6, 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. Step identifiers and dependenciesRule: Series_GENERAL_geometric_partial_sum_limit_and_upper_comparison · Uses: G001, S005 ## Reverse checks - 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. ## Result (6, 7, 1/7, 7, 49/6, Yes, upper_comparison_proves_convergence) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## 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: 9893cd411395199b7d2b795bdd7152f26e0cb83661028f6e5150149c45d3e9fd - candidate record SHA-256: d6f4e966dc5b477831cd4a62557395d46d305ff4228a2a387471f1af53aaa25b - semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252 - question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261117-r01-series-comparison-test-0006.json)