Question 23877: Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Original question and givens

Use mode=infinite_radius, positive scale=2, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).

• G001 — Use mode=infinite_radius, positive scale=2, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, 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-zero-infinity-prefix-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[mode, scale, coefficient_law, radius, symbolic_certificate, certificate]

ORIGINAL mode=infinite_radius,positive scale=2;unused-field sentinels prefix0,ratios1/1

WHOLE a_n=scale/n! for EVERY n>=0;for any fixed finite displacement t,absolute term ratio=|t|/(n+1) tends0,so absolute convergence at EVERY t;R=infinity

Reverse checks

• 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

(infinite_radius, 2, a_n=2/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)

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.