# Question 23813: The Positive and Negative Endpoints Can Have Different Status under the Same Radius

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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.

R0046-23813 · q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0063



## Original question and givens

Use a_n=3^(−n)/(n+1)^0 for every n≥0 and center=0, with radius R=3>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).


- G001 — Use a_n=3^(−n)/(n+1)^0 for every n≥0 and center=0, with radius R=3>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).





## 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-endpoint-independent;original snapshot and mathematics retained

After

explicit ordered source output fields=[coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary]

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 · impliesBefore

ORIGINAL center=0;R=3>0;power=0;WHOLE a_n=R^(-n)/(n+1)^power for EVERY n>=0

After

coefficient ratio tends1/R,so radius=3;at c+R terms=1/(n+1)^0;at c-R terms=(-1)^n/(n+1)^0

Conditions: none

The whole law gives the ratio limit and radius;substitute at each endpoint separately,not a root-test limit guessed from a prefix.

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





S003 · impliesBefore

ORIGINAL endpoint power=0;positive endpoint=c+R,negative endpoint=c-R

After

positive_endpoint_status=divergence;negative_endpoint_status=divergence_terms_do_not_tend_to_zero;separate_endpoint_certificate=Yes

Conditions: none

For p0,terms fail to tend to0 at either endpoint.For p1,the positive harmonic series diverges while the alternating series converges conditionally.For p2,both converge absolutely by the p-series test.The radius alone determines none of these endpoint outcomes.

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









## Reverse checks


- 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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=3^(-n)/(n+1)^0_for_every_n>=0, 0, 3, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)

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: 0001f5acdb52cabb32983874ede0131174a695761c2c357f3a82c9248688c66f
- candidate record SHA-256: c8610d77a4e4738ff38b985ded2fa71229b2ec0e3e6419fe53ab58a9fb7f4475
- 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-0063.json)