Question 26975: A Positive Constant Lower Bound Gives Unbounded Partial Sums

Original question and givens

Use exact scales A=50,B=25. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_certificate, certificate).

• G001 — Use exact scales A=50,B=25. This family uses a_n=A≥b_n=B>0 for every n, rather than reversing the conclusion of an upper comparison. Give (a_scale, b_scale, term_lower_bound, unbounded_partial_sum_statement, divergence_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-series-comparison-test

Step-by-step derivation

source roles = [series-lower-divergence]

derived = [[a_scale, b_scale, whole_term_bound, partial_sums, unbounded, certificate]]

source roles = [50, 25]

derived = [a_n=A,b_n=B for EVERY n>=0, S_N(a)=N*A>=N*B, B>0 => for every M choose N>M/B, Yes]

Reverse checks

• S002 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
• S001 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.

Result

(50, 25, a_n>=b_n>0_for_all_n, partial_sum_b=N*b_unbounded, Yes, lower_comparison_proves_divergence)

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.