Question 23859: Classify the Exact Distance from the Declared Center

Original question and givens

Use a_n=(9/10)^n for every n≥0, positive q=9/10, center=-4, point=-4. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).

• G001 — Use a_n=(9/10)^n for every n≥0, positive q=9/10, center=-4, point=-4. 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.

• mko-power-series-radius-of-convergence

Step-by-step derivation

source operation=radius-inside-outside-audit;original snapshot and mathematics retained

explicit ordered source output fields=[coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate]

Radius ORIGINAL positive coefficient-ratio reciprocal := div(1, 9/10)

Radius ORIGINAL positive coefficient-ratio reciprocal = 10/9

ORIGINAL WHOLE a_n=(9/10)^n for EVERY n>=0;center=-4;positive q=9/10

absolute geometric term ratio=q*|x-center|;less than1 gives absolute convergence,greater than1 makes terms not tend0;exact radius1/q=10/9

Radius ORIGINAL point minus SAME center := sub(-4, -4)

Radius ORIGINAL point minus SAME center = 0

ORIGINAL point=-4,SAME center=-4;signed difference=0;absolute distance=0;radius=10/9

distance<radius=Yes;classification=absolute_convergence;distance equality is OUTSIDE this operation,not silently classified

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 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=(9/10)^n_for_every_n>=0, -4, -4, 0, 10/9, absolute_convergence, Yes, 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.