# Question 23852: Classify the Exact Distance from the Declared Center

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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.

R0046-23852 · q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0027



## Original question and givens

Use a_n=(2/3)^n for every n≥0, positive q=2/3, center=-11, point=-9. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).


- G001 — Use a_n=(2/3)^n for every n≥0, positive q=2/3, center=-11, point=-9. 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](https://ocme.evemisslab.com/en/mko/mko-power-series-radius-of-convergence/)





## Step-by-step derivation

S001 · impliesBefore

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.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · calculationBefore

Radius ORIGINAL positive coefficient-ratio reciprocal := div(1, 2/3)

After

Radius ORIGINAL positive coefficient-ratio reciprocal = 3/2

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

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





S003 · impliesBefore

ORIGINAL WHOLE a_n=(2/3)^n for EVERY n>=0;center=-11;positive q=2/3

After

absolute geometric term ratio=q*|x-center|;less than1 gives absolute convergence,greater than1 makes terms not tend0;exact radius1/q=3/2

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.

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





S004 · calculationBefore

Radius ORIGINAL point minus SAME center := sub(-9, -11)

After

Radius ORIGINAL point minus SAME center = 2

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

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





S005 · impliesBefore

ORIGINAL point=-9,SAME center=-11;signed difference=2;absolute distance=2;radius=3/2

After

distance<radius=No;classification=divergence;distance equality is OUTSIDE this operation,not silently classified

Conditions: none

Use absolute distance from the same center,not the signed point.Equality with R belongs to the separate endpoint branch,not inside or outside.

Step identifiers and dependenciesRule: radius_exact_distance_center_binding_excludes_ENDPOINT_equality · 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 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=(2/3)^n_for_every_n>=0, -11, -9, 2, 3/2, 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 13a95246e9fdad8c1d8b39fd06e74c6907e6b87d7a0c4a7c99f5b5e2889f7d4f
- candidate record SHA-256: 1688c30f137ae1b86950ae6cde50af6ac5fcea174e15938aeab07cb1093910aa
- semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
- question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0027.json)