Question 189: A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit

Original question and givens

Use exact scales A=14,B=14, rational ratio r=4/10. Set A=14>0,r=4/10∈(0,1), shift=14, alternating=true. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).

• G001 — Use exact scales A=14,B=14, rational ratio r=4/10. Set A=14>0,r=4/10∈(0,1), shift=14, alternating=true. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).

Definition and scope

Require A>0, 0<r<1, s≥1 and n≥0. The explicit error bound, induction and geometric bound apply to the entire tail.

• mko-ratio-test-series

Step-by-step derivation

a_n=14*(-1)^n*(2/5)^n/(n+14)

|a_n|=14·(2/5)^n/(n+14)>0 for every integer n≥0

|a_n|=14·(2/5)^n/(n+14)

|a_(n+1)|=14·(2/5)^(n+1)/(n+15)

|a_(n+1)| / |a_n|

abs_ratio=2/5*(n+14)/(n+15)

abs_ratio=2/5*(n+14)/(n+15)

R_n=2/5−(2/5)/(n+15); |R_n−r|=(2/5)/(n+15)

epsilon>0; N=floor(r/epsilon)+1

n≥N ⇒ n+15>r/epsilon ⇒ |R_n−r|<epsilon

q−r := sub(7/10, 2/5)

q−r = 3/10

1−q := sub(1, 7/10)

1−q = 3/10

|a_0| := div(14, 14)

|a_0| = 1

0<R_n<r=2/5<q=7/10<1

|a_n|≤(1)·(7/10)^n for every n≥0

sum from k=0 to M of |a_k| ≤ (1)·(1−(7/10)^(M+1))/(1−(7/10))

all absolute partial sums ≤ 10/3

M>N≥0

sum from k=N+1 to M of |a_k| ≤ (1)·(7/10)^(N+1)/(1−(7/10)) → 0

Reverse checks

• S011 · premise_and_result_audit — The conclusion holds with or without alternating signs; no unrequested exact series sum is claimed.
• S010 · premise_and_result_audit — Monotone bounded convergence and the geometric-sum identity are explicit prerequisites, not a claim of a Lean proof.
• S009 · premise_and_result_audit — This is an explicit induction argument, not an inference from finitely many checked ratios.
• S008 · premise_and_result_audit — Recompute |a_0| with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute 1−q with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute q−r with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — N depends only on epsilon and the fixed parameters, not on a later n.
• S004 · premise_and_result_audit — The error tends to zero while the denominator grows.
• S003 · premise_and_result_audit — Keep the absolute ratio, not a potentially negative signed ratio.
• S002 · premise_and_result_audit — Do not shift the exponent while leaving the denominator unchanged.
• S001 · premise_and_result_audit — Positive A, r and n+s ensure every following ratio denominator is nonzero.

Result

(14, 2/5, 14, Yes, a_n=14*(-1)^n*(2/5)^n/(n+14), abs_ratio=2/5*(n+14)/(n+15), limit=2/5, 7/10, Yes, eventual_geometric_domination, absolutely_convergent)

The series converges absolutely, with an explicit geometric tail bound.

This question review and source binding

Review date: 2026-10-24

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.