Question 23809: The Positive and Negative Endpoints Can Have Different Status under the Same Radius

Original question and givens

Use a_n=4^(−n)/(n+1)^2 for every n≥0 and center=-4, with radius R=4>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=4^(−n)/(n+1)^2 for every n≥0 and center=-4, with radius R=4>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

Step-by-step derivation

source operation=radius-endpoint-independent;original snapshot and mathematics retained

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

ORIGINAL center=-4;R=4>0;power=2;WHOLE a_n=R^(-n)/(n+1)^power for EVERY n>=0

coefficient ratio tends1/R,so radius=4;at c+R terms=1/(n+1)^2;at c-R terms=(-1)^n/(n+1)^2

ORIGINAL endpoint power=2;positive endpoint=c+R,negative endpoint=c-R

positive_endpoint_status=absolute_convergence;negative_endpoint_status=absolute_convergence;separate_endpoint_certificate=Yes

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=4^(-n)/(n+1)^2_for_every_n>=0, -4, 4, 2, absolute_convergence, absolute_convergence, 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.