Explore related knowledge and questions →
← Question derivation index · Original question batch
R0048-24654 · q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0054
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).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 = [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.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
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.
Rule: OGF_WHOLE_geometric_limit_with_analytic_domain · Uses: G001, S004
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.(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.
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: 8cdc268857183d3191d2fa418fb1da065c83170bdce4824d1d647e3e54597241candidate record SHA-256: 45d311ece4913a70fa409cdc187c72b867faaad7e28dc44cae414227d78b0c65semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9Lean: NOT_RUN · human review: NOT_PERFORMED