# Question 24654: The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One

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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.

R0048-24654 · q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0054



## Original question and givens

Use a_n=(4/5)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).


- G001 — Use a_n=(4/5)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, 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-analytic-ordinary-generating-function](https://ocme.evemisslab.com/en/mko/mko-analytic-ordinary-generating-function/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [ogf-geometric-closed-form]

After

derived = [[coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate]]

Conditions: none

List the full source field order without JavaScript,keeping original data,Boolean flags,mathematical objects and nonclaims distinct.

Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001





S002 · calculationBefore

OGF ORIGINAL ratio times x := mul(4/5, 1/2)

After

OGF ORIGINAL ratio times x = 2/5

Conditions: none

Substitute original roles and established intermediates exactly,checking nonzero denominators without floats.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

OGF geometric denominator := sub(1, 2/5)

After

OGF geometric denominator = 3/5

Conditions: none

Substitute original roles and established intermediates exactly,checking nonzero denominators without floats.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

OGF geometric analytic value := div(1, 3/5)

After

OGF geometric analytic value = 5/3

Conditions: none

Substitute original roles and established intermediates exactly,checking nonzero denominators without floats.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · impliesBefore

source roles = [4/5, 1/2, 2/5]

After

derived = [Yes, 5/3, for EVERY N,S_N=(1-z^(N+1))/(1-z);z^(N+1)->0]

Conditions: none

The whole law a_n=r^n is declared.|rx|<1 gives a nonzero denominator and vanishing geometric tail,justifying analytic equality.A nonzero denominator outside convergence is insufficient.

Step identifiers and dependenciesRule: OGF_WHOLE_geometric_limit_with_analytic_domain · 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 — Recompute OGF geometric analytic value with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute OGF geometric denominator with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute OGF ORIGINAL ratio times x with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(a_n=(4/5)^n_for_every_n>=0, 1/2, 2/5, 5/3, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)

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: 8cdc268857183d3191d2fa418fb1da065c83170bdce4824d1d647e3e54597241
- candidate record SHA-256: 45d311ece4913a70fa409cdc187c72b867faaad7e28dc44cae414227d78b0c65
- semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0
- question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0054.json)