Question 23878: Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii

Original question and givens

Use mode=finite_prefix, positive scale=3, source prefix_length=1, first_ratio=1/2, second_ratio=1/3. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate).

• G001 — Use mode=finite_prefix, positive scale=3, source prefix_length=1, first_ratio=1/2, second_ratio=1/3. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, 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, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate]

Radius ORIGINAL first complete-tail reciprocal := div(1, 1/2)

Radius ORIGINAL first complete-tail reciprocal = 2

Radius ORIGINAL second complete-tail reciprocal := div(1, 1/3)

Radius ORIGINAL second complete-tail reciprocal = 3

ORIGINAL prefix_length parameter N=1;positive scale=3;distinct positive tail ratios=[1/2, 1/3]

COMMON zero coefficients at ALL n0..N=[0, 0];there are N+1=2 zeros;for n>N laws are scale*q^(n-N-1),first tail term scale;radii=[2, 3] DIFFER

Reverse checks

• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Recompute Radius ORIGINAL second complete-tail reciprocal with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Radius ORIGINAL first complete-tail 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

(finite_prefix, 3, 1, [0, 0], a_n=0_for_n<=1;a_n=3*(1/2)^(n-2)_for_n>1, b_n=0_for_n<=1;b_n=3*(1/3)^(n-2)_for_n>1, 2, 3, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)

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.