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

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

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

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

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

positive_endpoint_status=divergence;negative_endpoint_status=divergence_terms_do_not_tend_to_zero;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=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.