# Question 17901: A Finite Sum of Absolute Values Is Not an Infinite Certificate [Return to catalogue page 90](https://ocme.evemisslab.com/en/questions/page/90/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations/#q-20261014-advance-20261104-r03-absolute-convergence-of-series-0051) 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. R0034-17901 · q-20261014-advance-20261104-r03-absolute-convergence-of-series-0051 ## Original question and givens Supplied ordered exact rational terms=[1/2, -1/3]. Give (terms, absolute_terms, absolute_partial_sum, boundary). - G001 — Supplied ordered exact rational terms=[1/2, -1/3]. Give (terms, absolute_terms, absolute_partial_sum, boundary). ## 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-absolute-convergence-of-series](https://ocme.evemisslab.com/en/mko/mko-absolute-convergence-of-series/) ## Step-by-step derivation S001 · impliesBefore source operation=absseries-absolute-partial;original question snapshot retained;source presentation interface resolved in this page After EXPLICIT ordered output fields=[terms, absolute_terms, absolute_partial_sum, boundary] Conditions: none Preserve the original snapshot's historical interface marker while listing the full ordered fields here,without browser JavaScript or an external lookup.Field labels are presentation metadata,not mathematical evaluation.The fourth term-zero No marks insufficiency,not a mislabeled second convergence flag. Step identifiers and dependenciesRule: absseries_explicit_source_interface_fields_without_javascript · Uses: G001 S002 · impliesBefore original ordered signed term index=0;value=1/2 After absolute term at SAME index=1/2 Conditions: none Take the absolute value of each original term in place;do not cancel signs first and take only the absolute value of the total. Step identifiers and dependenciesRule: absseries_each_term_absolute_before_sum · Uses: G001, S001 S003 · calculationBefore finite absolute term1 sign reversal := mul(-1, -1/3) After finite absolute term1 sign reversal = 1/3 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore original ordered signed term index=1;value=-1/3 After absolute term at SAME index=1/3 Conditions: none Take the absolute value of each original term in place;do not cancel signs first and take only the absolute value of the total. Step identifiers and dependenciesRule: absseries_each_term_absolute_before_sum · Uses: G001, S003 S005 · calculationBefore finite absolute partial sum := sum(1/2, 1/3) After finite absolute partial sum = 5/6 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore finite original signed sum for distinction := sum(1/2, -1/3) After finite original signed sum for distinction = 1/6 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore finite signed terms=[1/2, -1/3];same-order absolute terms=[1/2, 1/3] After sum absolute terms=5/6;absolute signed sum=1/6<=that;finite_absolute_partial_sum_only Conditions: none These quantities may differ;both empty sums are0.This is the supplied finite data,not an absolute-convergence certificate for an unspecified infinite continuation. Step identifiers and dependenciesRule: absseries_finite_absolute_sum_not_absolute_signed_sum_or_infinite_tail · Uses: G001, S006 ## Reverse checks - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute finite original signed sum for distinction with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute finite absolute partial sum with exact rational arithmetic, not rounded decimals. - 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 finite absolute term1 sign reversal with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([1/2, -1/3], [1/2, 1/3], 5/6, finite_absolute_partial_sum_only) 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: 6fda49737532fb177701834d197294d71b79b9d8c465c7c3210a59a4e7f17a38 - candidate record SHA-256: 2134d2dcd8c9a3f762c4269b6e647c030380596030b674fbeaa48428aed1082a - semantic review receipt SHA-256: f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336 - question-first witness SHA-256: 22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0051.json)