# Question 17782: Terms Tending to Zero Do Not Make a Harmonic Tail Summable [Return to catalogue page 89](https://ocme.evemisslab.com/en/questions/page/89/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261103-r03-infinite-series-foundations/#q-20261014-advance-20261103-r03-infinite-series-0082) 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. R0034-17782 · q-20261014-advance-20261103-r03-infinite-series-0082 ## Original question and givens Use the shifted harmonic terms 1/(n+shift), n=0,1,..., with positive shift=7 and displayed prefix_length=5. Give (shift,prefix_length,terms,terms_tend_zero?,series_converges?,terms_tend_to_zero_but_shifted_harmonic_series_diverges). - G001 — Use the shifted harmonic terms 1/(n+shift), n=0,1,..., with positive shift=7 and displayed prefix_length=5. Give (shift,prefix_length,terms,terms_tend_zero?,series_converges?,terms_tend_to_zero_but_shifted_harmonic_series_diverges). ## 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-infinite-series](https://ocme.evemisslab.com/en/mko/mko-infinite-series/) ## Step-by-step derivation S001 · calculationBefore harmonic displayed index0 denominator := sum(0, 7) After harmonic displayed index0 denominator = 7 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 · calculationBefore harmonic displayed index0 term := div(1, 7) After harmonic displayed index0 term = 1/7 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 harmonic displayed index1 denominator := sum(1, 7) After harmonic displayed index1 denominator = 8 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 harmonic displayed index1 term := div(1, 8) After harmonic displayed index1 term = 1/8 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 harmonic displayed index2 denominator := sum(2, 7) After harmonic displayed index2 denominator = 9 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 harmonic displayed index2 term := div(1, 9) After harmonic displayed index2 term = 1/9 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 harmonic displayed index3 denominator := sum(3, 7) After harmonic displayed index3 denominator = 10 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 harmonic displayed index3 term := div(1, 10) After harmonic displayed index3 term = 1/10 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 harmonic displayed index4 denominator := sum(4, 7) After harmonic displayed index4 denominator = 11 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 harmonic displayed index4 term := div(1, 11) After harmonic displayed index4 term = 1/11 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 · impliesBefore shifted harmonic series:DECLARED whole positive tail b_k=1/(k+7),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, S011 S013 · impliesBefore displayed finite prefix=[1/7, 1/8, 1/9, 1/10, 1/11];SEPARATE complete harmonic-tail proof retained After terms tend to0=Yes;series converges=No;terms_tend_to_zero_but_shifted_harmonic_series_diverges Conditions: none The finite list is illustrative.The two infinite claims come separately from the declared full tail and block argument;the necessary term-zero condition is not sufficient. Step identifiers and dependenciesRule: series_term_limit_not_partial_sum_limit · Uses: G001, S012 ## Reverse checks - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Recompute harmonic displayed index4 term with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute harmonic displayed index4 denominator with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute harmonic displayed index3 term with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute harmonic displayed index3 denominator with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute harmonic displayed index2 term with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute harmonic displayed index2 denominator with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute harmonic displayed index1 term with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute harmonic displayed index1 denominator with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute harmonic displayed index0 term with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Recompute harmonic displayed index0 denominator with exact rational arithmetic, not rounded decimals. ## Result (7, 5, [1/7, 1/8, 1/9, 1/10, 1/11], Yes, No, terms_tend_to_zero_but_shifted_harmonic_series_diverges) 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: c5a4452ea7d82ba7773b4f53595146f9358c9152b9f51760ec79349f478d4df5 - candidate record SHA-256: 726ab05aed9e28baa93f6478f2dc4469fb3e4f91997e4e392402c77ca76ba257 - 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-20261103-r03-infinite-series-0082.json)