# Question 17948: Conditional Convergence Needs Both Convergence and Absolute Divergence [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-0048) 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. R0034-17948 · q-20261014-advance-20261104-r03-absolute-convergence-of-series-0048 ## Original question and givens Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=23, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate). - G001 — Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=23, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, 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-absolute-convergence-of-series](https://ocme.evemisslab.com/en/mko/mko-absolute-convergence-of-series/) ## Step-by-step derivation S001 · impliesBefore source operation=absseries-conditional-alternating;original question snapshot retained;source presentation interface resolved in this page After EXPLICIT ordered output fields=[shift, n, terms, series_converges, absolutely_converges, certificate] 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 · calculationBefore alternating displayed term0 denominator := sum(0, 23) After alternating displayed term0 denominator = 23 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, S001 S003 · calculationBefore alternating displayed term0 signed value := div(1, 23) After alternating displayed term0 signed value = 1/23 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 · calculationBefore alternating displayed term1 denominator := sum(1, 23) After alternating displayed term1 denominator = 24 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, S003 S005 · calculationBefore alternating displayed term1 signed value := div(-1, 24) After alternating displayed term1 signed value = -1/24 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 alternating displayed term2 denominator := sum(2, 23) After alternating displayed term2 denominator = 25 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 · calculationBefore alternating displayed term2 signed value := div(1, 25) After alternating displayed term2 signed value = 1/25 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, S006 S008 · calculationBefore alternating displayed term3 denominator := sum(3, 23) After alternating displayed term3 denominator = 26 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, S007 S009 · calculationBefore alternating displayed term3 signed value := div(-1, 26) After alternating displayed term3 signed value = -1/26 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, S008 S010 · calculationBefore alternating displayed term4 denominator := sum(4, 23) After alternating displayed term4 denominator = 27 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, S009 S011 · calculationBefore alternating displayed term4 signed value := div(1, 27) After alternating displayed term4 signed value = 1/27 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, S010 S012 · impliesBefore absolute series of declared alternating tail:DECLARED whole positive tail b_k=1/(k+23),k>=0;shift>0 After b_k decreases to0:for ANY epsilon>0 choose N with N+shift>1/epsilon,then every k>=N has b_k=0 large enough that2^j>=shift-1,the block m=2^j+1..2^(j+1) has2^j terms each>=1/2^(j+1);each block sum>=1/2;infinitely many disjoint blocks make partial sums unbounded Conditions: none Removing finitely many initial terms leaves the dyadic-block argument.Each sufficiently late block contributes at least1/2,so arbitrarily many disjoint blocks force unbounded partial sums.Terms tending to zero do not give a finite limit of partial sums. Step identifiers and dependenciesRule: harmonic_shifted_dyadic_blocks_diverge_not_term_limit · Uses: G001, S012 S014 · impliesBefore WHOLE declared terms=(-1)^k/(k+23),k starts0 positive;displayed first5 terms=[1/23, -1/24, 1/25, -1/26, 1/27] After b_k positive decreasing to0;even S_2m increases by b_2m-b_(2m+1)>=0;odd S_(2m+1) decreases;0<=S_2m<=S_(2m+1)<=b_0=1/23;gap b_2m tends0 Conditions: none Positive decreasing magnitudes make even partial sums increase and odd partial sums decrease while bounding one another.Real completeness gives both limits,and their vanishing gap makes them equal,so the whole partial-sum sequence converges.Alternation or a term-zero condition alone is not a general criterion. Step identifiers and dependenciesRule: absseries_alternating_monotone_subsequences_not_alternation_alone · Uses: G001, S013 S015 · calculationBefore alternating displayed exact partial sum := sum(1/23, -1/24, 1/25, -1/26, 1/27) After alternating displayed exact partial sum = 65209/1614600 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, S014 S016 · calculationBefore alternating next-term remainder denominator := sum(5, 23) After alternating next-term remainder denominator = 28 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, S015 S017 · calculationBefore alternating next-term absolute remainder bound := div(1, 28) After alternating next-term absolute remainder bound = 1/28 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, S016 S018 · calculationBefore alternating next signed term := mul(-1, 1/28) After alternating next signed term = -1/28 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, S017 S019 · calculationBefore alternating next exact partial sum := sum(65209/1614600, -1/28) After alternating next exact partial sum = 52813/11302200 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, S018 S020 · impliesBefore displayed S_n=65209/1614600;adjacent S_(n+1)=52813/11302200;n=5;positive shift=23 After common limit L lies between these two endpoints;abs(L-S_n)<=1/28;original series converges=Yes;absolute harmonic series converges=No Conditions: none The whole-tail alternating certificate traps the common limit between adjacent partial sums and gives the next-term error bound.The separate dyadic argument proves divergence of the absolute series.Both certificates establish conditional convergence;the displayed finite list alone proves neither infinite claim. Step identifiers and dependenciesRule: absseries_conditional_requires_convergence_and_absolute_divergence · Uses: G001, S019 ## Reverse checks - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Recompute alternating next exact partial sum with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute alternating next signed term with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute alternating next-term absolute remainder bound with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute alternating next-term remainder denominator with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute alternating displayed exact partial sum with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute alternating displayed term4 signed value with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute alternating displayed term4 denominator with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute alternating displayed term3 signed value with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute alternating displayed term3 denominator with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute alternating displayed term2 signed value with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute alternating displayed term2 denominator with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute alternating displayed term1 signed value with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute alternating displayed term1 denominator with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute alternating displayed term0 signed value with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute alternating displayed term0 denominator 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 (23, 5, [1/23, -1/24, 1/25, -1/26, 1/27], Yes, No, alternating_remainder_and_harmonic_absolute_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: 8193b0a60142e3e74774537291f1941306b8e4b29faa344dfe29e24611f2a872 - candidate record SHA-256: 73b9cae21f3a34738d07aef59165eb08230cdb0cf7befe2cef3b2d74639ebb1d - 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-0048.json)