# Question 750: Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence

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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.

P1000-0750 · q-20261015-advance-20261117-r02-ratio-test-series-0075



## Original question and givens

Use a_n=1/(n+shift)^p with p=1∈{1,2}, shift=25≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).


- G001 — Use a_n=1/(n+shift)^p with p=1∈{1,2}, shift=25≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).





## Definition and scope

For p=1 and p=2 the ratio limit is 1; the classification comes from harmonic blocks and a telescoping bound, so the ratio test is inconclusive.


- [mko-ratio-test-series](https://ocme.evemisslab.com/en/mko/mko-ratio-test-series/)





## Step-by-step derivation

S001 · impliesBefore

a_n=1/(n+25)^1, n≥0

After

|a_(n+1)/a_n|=((n+25)/(n+26))^1

Conditions: none

Derive the absolute ratio from the full positive-term formula.

Step identifiers and dependenciesRule: successive_ratio · Uses: G001





S002 · impliesBefore

(n+25)/(n+26)=1−1/(n+26)

After

ratio limit=1

Conditions: none

The growing denominator makes the relative error vanish; the denominator itself does not vanish.

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





S003 · impliesBefore

k=n+s; block k=s·2^j through s·2^(j+1)−1

After

s·2^j terms, each >1/(s·2^(j+1))

Conditions: none

Each complete dyadic block has an explicit term count and denominator bound.

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





S004 · impliesBefore

block sum > (s·2^j)/(s·2^(j+1))

After

every block sum >1/2; J blocks sum >J/2

Conditions: none

Arbitrarily many blocks make partial sums unbounded; a finite starting shift s does not change the conclusion.

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





S005 · impliesBefore

same ratio limit1 for both exponent families

After

L=1 alone does not decide convergence

Conditions: none

The classification comes from the separate complete-tail arguments above, not from the ratio limit1.

Step identifiers and dependenciesRule: inconclusive_ratio_boundary · Uses: G001, S004









## Reverse checks


- S005 · premise_and_result_audit — Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values.
- S004 · premise_and_result_audit — Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values.
- S003 · premise_and_result_audit — Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values.
- S002 · premise_and_result_audit — Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values.
- S001 · premise_and_result_audit — Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values.





## Result

(1, 25, abs_ratio=((n+25)/(n+26))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.





## 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: cf29d8883ba4b2b839c278df7db3a31c52361b86283205863badf933dca6b7d2
- candidate record SHA-256: be8483d61f13c23fb5ccebdf4ead84051e542f561c2a893a3b0a888dba19320b
- semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772
- question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b
- Lean: NOT_RUN · human review: NOT_PERFORMED

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