S001 · implies
Before
a_n=1/(n+1)^1, n≥0
After
|a_(n+1)/a_n|=((n+1)/(n+2))^1
Conditions: none
Derive the absolute ratio from the full positive-term formula.
Step identifiers and dependencies
Rule: successive_ratio · Uses: G001
Explore related knowledge and questions →
Question 726: Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
On this page
← Question derivation index · Original question batch
P1000-0726 · q-20261015-advance-20261117-r02-ratio-test-series-0051
Original question and givens
Use a_n=1/(n+shift)^p with p=1∈{1,2}, shift=1≥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=1≥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+1)/(n+2)=1−1/(n+2)
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, 1, abs_ratio=((n+1)/(n+2))^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: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: e14bc1837d0454579ae680029edeb80f435071b1c108419181e00a0c17e6244fcandidate record SHA-256: 7a9382ce2648adfbd7932106898097c4aab9e4b7f58dc107489ba7bff4932d88semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612bLean: NOT_RUN · human review: NOT_PERFORMED