S001 · implies
Before
a_n=1/(n+4)^2, n≥0
After
|a_(n+1)/a_n|=((n+4)/(n+5))^2
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 729: Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
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P1000-0729 · q-20261015-advance-20261117-r02-ratio-test-series-0054
Original question and givens
Use a_n=1/(n+shift)^p with p=2∈{1,2}, shift=4≥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=2∈{1,2}, shift=4≥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+4)/(n+5)=1−1/(n+5)
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≥2
After
1/k² ≤1/(k(k−1)) =1/(k−1)−1/k
Conditions: none
For positive denominators k(k−1)≤k², and reciprocation gives the stated upper bound.
Step identifiers and dependencies
Rule: telescoping_majorant · Uses: G001, S002
S004 · implies
Before
sum k=K through M, K=max(s,2)
After
sum 1/k² ≤1/(K−1)−1/M ≤1/(K−1)
Conditions: none
The tail has a telescoping bound; if s=1 add the first term1. Positive partial sums are increasing and bounded, hence converge.
Step identifiers and dependencies
Rule: bounded_positive_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
(2, 4, abs_ratio=((n+4)/(n+5))^2, 1, convergent_p_series, 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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