# Question 26951: A Positive Constant Lower Bound Gives Unbounded Partial Sums

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0052-26951 · q-20261015-advance-20261117-r01-series-comparison-test-0026



## Original question and givens

Use exact scales A=2,B=1. 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=2,B=1. 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](https://ocme.evemisslab.com/en/mko/mko-series-comparison-test/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [series-lower-divergence]

After

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

Conditions: none

List this operation/mode exact source field order without JavaScript.Keep Boolean,string and null roles distinct;do not pad fields or infer the interface from answer shape.

Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001





S002 · impliesBefore

source roles = [2, 1]

After

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]

Conditions: none

Use the declared constant laws for every n,not sampled values.The N-term sum is at least N*B;since B>0,choose N>M/B for any threshold M,so the sums diverge to positive infinity.No extra bounded source multiplication is imposed.

Step identifiers and dependenciesRule: Series_WHOLE_constant_laws_and_arbitrary_N_unbounded_lower_sum · Uses: G001, S001









## 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

(2, 1, 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 3513c59b0e77cfaee050dc9ae4bfe0bdac54aa52b3af3764f9ed6d7e0f6f0b56
- candidate record SHA-256: f215195622d8d5d79af7fbf6f7dcaaacd89bafb2dc12630df51065090113a857
- semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252
- question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261117-r01-series-comparison-test-0026.json)