# Question 17984: Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion

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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.

R0034-17984 · q-20261014-advance-20261104-r03-absolute-convergence-of-series-0084



## Original question and givens

Use the shifted-harmonic boundary with positive shift=9 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=9 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.


- [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-term-zero-boundary;original question snapshot retained;source presentation interface resolved in this page

After

EXPLICIT ordered output fields=[shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, 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

absolute-convergence term-zero counterexample:DECLARED whole positive tail b_k=1/(k+9),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.

Step identifiers and dependenciesRule: harmonic_full_tail_terms_zero_not_observed_prefix · Uses: G001, S001





S003 · impliesBefore

absolute-convergence term-zero counterexample:same full absolute/positive series over denominators m=9,10,...

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.

Step identifiers and dependenciesRule: harmonic_shifted_dyadic_blocks_diverge_not_term_limit · Uses: G001, S002





S004 · impliesBefore

inspected finite prefix length=5;DECLARED harmonic tail shift=9;whole-tail term limit0 and divergent absolute sum proved separately

After

terms_tend_to_zero=Yes;term_zero_suffices_for_absolute_convergence=No;term_zero_not_absolute_convergence

Conditions: none

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.

Step identifiers and dependenciesRule: absseries_term_zero_insufficiency_marker_not_prefix_classification · Uses: G001, S003









## Reverse checks


- 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

(9, 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 1a661d93678148f1d12bf638f1daf1db1a133f30d16b34b00e3254cff7dabc9f
- candidate record SHA-256: e62098a66118c54cda89c1669ea5f3040d32bbc71be8cc43df47b6cbc03e757a
- 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-0084.json)