Question 24251: The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Original question and givens

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/2, N=2, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

• G001 — Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/2, N=2, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_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-real-exponential-function

Step-by-step derivation

source roles = [exp-infinite-radius-certificate]

derived = [[epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate]]

Exp tail coefficient ratio bound := div(1, 3)

Exp tail coefficient ratio bound = 1/3

Exp declared positive epsilon := div(1, 2)

Exp declared positive epsilon = 1/2

source roles = [2, 2, 1/3, 1/2]

derived = [EVERY n>=N:1/(n+1)<=1/(N+1)<1/d, for ANY epsilon>0 choose integer N with N+1>1/epsilon, coefficient ratio limit0,radius infinity]

source roles = [[2, 2], a_n=1/n! for EVERY n>=0]

derived = [coefficient ratio=1/(n+1), for EVERY fixed real z,term absolute ratio=|z|/(n+1)->0, absolute convergence and real-valued E(z)]

Reverse checks

• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 Exp declared positive epsilon with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Exp tail coefficient ratio bound 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

(2, 2, epsilon=1/2, |a_(n+1)/a_n|=1/(n+1), 1/3, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

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.