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R0034-17782 · q-20261014-advance-20261103-r03-infinite-series-0082
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).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.
Before
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.
Rule: exact_rational_arithmetic · Uses: G001
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
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<epsilon
Conditions: none
Increasing denominators and an Archimedean cutoff supply the all-later bound for every epsilon,not an inference from the displayed prefix.
Rule: harmonic_full_tail_terms_zero_not_observed_prefix · Uses: G001, S010
Before
shifted harmonic series:same full absolute/positive series over denominators m=7,8,...
After
for every integer j>=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.
Rule: harmonic_shifted_dyadic_blocks_diverge_not_term_limit · Uses: G001, S011
Before
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.
Rule: series_term_limit_not_partial_sum_limit · Uses: G001, S012
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.(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.
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.
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