Use a_n=(16/17)^n for every n≥0, positive q=16/17, center=3, point=5. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).
G001 — Use a_n=(16/17)^n for every n≥0, positive q=16/17, center=3, point=5. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, 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.
source operation=radius-inside-outside-audit;original snapshot and mathematics retained
After
explicit ordered source output fields=[coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate]
Conditions: none
Keep the source snapshot while listing ordered fields without JavaScript.Radius uses its original mode to select6 or11 fields without padding;distinguish nonclaims from actual negations.
ORIGINAL WHOLE a_n=(16/17)^n for EVERY n>=0;center=3;positive q=16/17
After
absolute geometric term ratio=q*|x-center|;less than1 gives absolute convergence,greater than1 makes terms not tend0;exact radius1/q=17/16
Conditions: none
Use the law for every n,not a handful of coefficients.Keep interior and exterior separate and audit endpoints separately;a radius does not replace endpoint conclusions.
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 Radius ORIGINAL point minus SAME center with exact rational arithmetic, not rounded decimals.
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 Radius ORIGINAL positive coefficient-ratio reciprocal 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=(16/17)^n_for_every_n>=0, 3, 5, 2, 17/16, divergence, No, classification_uses_exact_distance_from_center)
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.