S001 · implies
Before
a_n=1/(n+7)^1, n≥0
After
|a_(n+1)/a_n|=((n+7)/(n+8))^1
Conditions: none
Derive the absolute ratio from the full positive-term formula.
Step identifiers and dependencies
Rule: successive_ratio · Uses: G001
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Question 732: Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
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P1000-0732 · q-20261015-advance-20261117-r02-ratio-test-series-0057
Original question and givens
Use a_n=1/(n+shift)^p with p=1∈{1,2}, shift=7≥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=7≥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.
Step-by-step derivation
S002 · implies
Before
(n+7)/(n+8)=1−1/(n+8)
After
ratio limit=1
Conditions: none
The growing denominator makes the relative error vanish; the denominator itself does not vanish.
Step identifiers and dependencies
Rule: rational_ratio_limit · Uses: G001, S001
S003 · implies
Before
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 dependencies
Rule: harmonic_block_count · Uses: G001, S002
S004 · implies
Before
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 dependencies
Rule: unbounded_harmonic_partials · Uses: G001, S003
S005 · implies
Before
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 dependencies
Rule: 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, 7, abs_ratio=((n+7)/(n+8))^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
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