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Question 17720: The Geometric-Series Formula Retains Its Absolute-Ratio Premise

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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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.

Step-by-step derivation

S001 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001

S002 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

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 dependencies

Rule: series_geometric_exact_input_roles_and_absolute_ratio_domain · Uses: G001, S002

S004 · implies

Before

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 dependencies

Rule: series_geometric_all_N_finite_cancellation_identity · Uses: G001, S003

S005 · implies

Before

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 dependencies

Rule: 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
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