Use the shifted-harmonic boundary with positive shift=10 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).
G001 — Use the shifted-harmonic boundary with positive shift=10 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, 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.
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.
absolute-convergence term-zero counterexample:same full absolute/positive series over denominators m=10,11,...
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.
The fourth No answers whether the term-zero condition suffices,not a convergence test inferred from this prefix.The complete harmonic counterexample has terms tending to0 while its absolute series diverges.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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
(10, 5, Yes, No, term_zero_not_absolute_convergence)
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.