# Question 17720: The Geometric-Series Formula Retains Its Absolute-Ratio Premise [Return to catalogue page 89](https://ocme.evemisslab.com/en/questions/page/89/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261103-r03-infinite-series-foundations/#q-20261014-advance-20261103-r03-infinite-series-0045) 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. R0034-17720 · q-20261014-advance-20261103-r03-infinite-series-0045 ## Original question and givens Use the infinite series with term a*r^k for k=0,1,..., exact a=20/21 and r=1/2, requiring |r|<1. Give (a,r,infinite_sum,absolute_ratio_below_one_symbolic_tail). - G001 — Use the infinite series with term a*r^k for k=0,1,..., exact a=20/21 and r=1/2, requiring |r|<1. Give (a,r,infinite_sum,absolute_ratio_below_one_symbolic_tail). ## 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-infinite-series](https://ocme.evemisslab.com/en/mko/mko-infinite-series/) ## Step-by-step derivation S001 · calculationBefore signed geometric series one minus effective ratio := sub(1, 1/2) After signed geometric series one minus effective ratio = 1/2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · calculationBefore signed geometric series exact infinite candidate sum := div(20/21, 1/2) After signed geometric series exact infinite candidate sum = 40/21 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore signed geometric series:original SIGNED a=20/21;SIGNED r=1/2;absolute ratio=1/2<1;effective terms=(20/21)*(1/2)^k,k=0,1,... After candidate sum=40/21;effective denominator1-ratio=1/2 is nonzero Conditions: none Retain signed a and r.Change to effective absolute values only when this operation asks for the absolute series.The |r|<1 guard is this certificate's domain;finite samples or an unrelated zero-coefficient exception do not change it. Step identifiers and dependenciesRule: series_geometric_exact_input_roles_and_absolute_ratio_domain · Uses: G001, S002 S004 · impliesBefore signed geometric series:for ANY N>=1,the first N terms start at k0 with leading=20/21,ratio=1/2 After (1-ratio)*S_N=(20/21)*(1-(1/2)^N);S_N=(20/21)*(1-(1/2)^N)/(1/2);candidate minus S_N=(20/21)*(1/2)^N/(1/2) Conditions: none Multiply the finite sum by1−ratio and cancel adjacent powers,leaving endpoints.This identity holds for arbitrary N≥1,not a formula guessed from several partial sums. Step identifiers and dependenciesRule: series_geometric_all_N_finite_cancellation_identity · Uses: G001, S003 S005 · impliesBefore signed geometric series:|effective ratio|=1/2<1;remainder magnitude=abs(20/21)*abs(1/2)^N/abs(1/2) After geometric power tends to0;remainder tends to0;partial sums converge to the exact candidate Conditions: none Powers of a ratio with absolute value strictly below1 tend to0.Fixed finite coefficients and a nonzero denominator preserve this remainder limit,which proves convergence of the partial-sum sequence. Step identifiers and dependenciesRule: series_geometric_remainder_limit_not_finite_prefix_inference · Uses: G001, S004 ## Reverse checks - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Recompute signed geometric series exact infinite candidate sum with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Recompute signed geometric series one minus effective ratio with exact rational arithmetic, not rounded decimals. ## Result (20/21, 1/2, 40/21, absolute_ratio_below_one_symbolic_tail) 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: a404aa167c1d3ccb06bb9f2a7cbac487bcb62ea571f9aea21eccbfea16c74d34 - candidate record SHA-256: 8cf13ec3a2edbc304d8253b71cd7251187f21b6cbe47f51157e50c094a7e4559 - semantic review receipt SHA-256: f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336 - question-first witness SHA-256: 22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261103-r03-infinite-series-0045.json)