Absolute Ratio Limits, Domination and Limit-One Inconclusiveness
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261015-advance-20261117-r02-ratio-test-series-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=1,B=1, rational ratio r=1/7. Set A=1>0,r=1/7∈(0,1), shift=1, alternating=false. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(1, 1/7, 1, No, a_n=1*(1/7)^n/(n+1), abs_ratio=1/7*(n+1)/(n+2), limit=1/7, 4/7, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (1, 1/7, 1, No, a_n=1*(1/7)^n/(n+1), abs_ratio=1/7*(n+1)/(n+2), limit=1/7, 4/7, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
2. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=2,B=2, rational ratio r=2/8. Set A=2>0,r=2/8∈(0,1), shift=2, alternating=true. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(2, 1/4, 2, Yes, a_n=2*(-1)^n*(1/4)^n/(n+2), abs_ratio=1/4*(n+2)/(n+3), limit=1/4, 5/8, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (2, 1/4, 2, Yes, a_n=2*(-1)^n*(1/4)^n/(n+2), abs_ratio=1/4*(n+2)/(n+3), limit=1/4, 5/8, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
3. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=3,B=3, rational ratio r=3/9. Set A=3>0,r=3/9∈(0,1), shift=3, alternating=false. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(3, 1/3, 3, No, a_n=3*(1/3)^n/(n+3), abs_ratio=1/3*(n+3)/(n+4), limit=1/3, 2/3, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (3, 1/3, 3, No, a_n=3*(1/3)^n/(n+3), abs_ratio=1/3*(n+3)/(n+4), limit=1/3, 2/3, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
4. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=4,B=4, rational ratio r=4/10. Set A=4>0,r=4/10∈(0,1), shift=4, alternating=true. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(4, 2/5, 4, Yes, a_n=4*(-1)^n*(2/5)^n/(n+4), abs_ratio=2/5*(n+4)/(n+5), limit=2/5, 7/10, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (4, 2/5, 4, Yes, a_n=4*(-1)^n*(2/5)^n/(n+4), abs_ratio=2/5*(n+4)/(n+5), limit=2/5, 7/10, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
5. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=5,B=5, rational ratio r=5/11. Set A=5>0,r=5/11∈(0,1), shift=5, alternating=false. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(5, 5/11, 5, No, a_n=5*(5/11)^n/(n+5), abs_ratio=5/11*(n+5)/(n+6), limit=5/11, 8/11, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (5, 5/11, 5, No, a_n=5*(5/11)^n/(n+5), abs_ratio=5/11*(n+5)/(n+6), limit=5/11, 8/11, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
6. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=6,B=6, rational ratio r=1/7. Set A=6>0,r=1/7∈(0,1), shift=6, alternating=true. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(6, 1/7, 6, Yes, a_n=6*(-1)^n*(1/7)^n/(n+6), abs_ratio=1/7*(n+6)/(n+7), limit=1/7, 4/7, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (6, 1/7, 6, Yes, a_n=6*(-1)^n*(1/7)^n/(n+6), abs_ratio=1/7*(n+6)/(n+7), limit=1/7, 4/7, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
7. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=7,B=7, rational ratio r=2/8. Set A=7>0,r=2/8∈(0,1), shift=7, alternating=false. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(7, 1/4, 7, No, a_n=7*(1/4)^n/(n+7), abs_ratio=1/4*(n+7)/(n+8), limit=1/4, 5/8, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (7, 1/4, 7, No, a_n=7*(1/4)^n/(n+7), abs_ratio=1/4*(n+7)/(n+8), limit=1/4, 5/8, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
8. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=8,B=8, rational ratio r=3/9. Set A=8>0,r=3/9∈(0,1), shift=8, alternating=true. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(8, 1/3, 8, Yes, a_n=8*(-1)^n*(1/3)^n/(n+8), abs_ratio=1/3*(n+8)/(n+9), limit=1/3, 2/3, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (8, 1/3, 8, Yes, a_n=8*(-1)^n*(1/3)^n/(n+8), abs_ratio=1/3*(n+8)/(n+9), limit=1/3, 2/3, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
9. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=9,B=9, rational ratio r=4/10. Set A=9>0,r=4/10∈(0,1), shift=9, alternating=false. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(9, 2/5, 9, No, a_n=9*(2/5)^n/(n+9), abs_ratio=2/5*(n+9)/(n+10), limit=2/5, 7/10, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (9, 2/5, 9, No, a_n=9*(2/5)^n/(n+9), abs_ratio=2/5*(n+9)/(n+10), limit=2/5, 7/10, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
10. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=10,B=10, rational ratio r=5/11. Set A=10>0,r=5/11∈(0,1), shift=10, alternating=true. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(10, 5/11, 10, Yes, a_n=10*(-1)^n*(5/11)^n/(n+10), abs_ratio=5/11*(n+10)/(n+11), limit=5/11, 8/11, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (10, 5/11, 10, Yes, a_n=10*(-1)^n*(5/11)^n/(n+10), abs_ratio=5/11*(n+10)/(n+11), limit=5/11, 8/11, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
11. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=11,B=11, rational ratio r=1/7. Set A=11>0,r=1/7∈(0,1), shift=11, alternating=false. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(11, 1/7, 11, No, a_n=11*(1/7)^n/(n+11), abs_ratio=1/7*(n+11)/(n+12), limit=1/7, 4/7, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (11, 1/7, 11, No, a_n=11*(1/7)^n/(n+11), abs_ratio=1/7*(n+11)/(n+12), limit=1/7, 4/7, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
12. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=12,B=12, rational ratio r=2/8. Set A=12>0,r=2/8∈(0,1), shift=12, alternating=true. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(12, 1/4, 12, Yes, a_n=12*(-1)^n*(1/4)^n/(n+12), abs_ratio=1/4*(n+12)/(n+13), limit=1/4, 5/8, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (12, 1/4, 12, Yes, a_n=12*(-1)^n*(1/4)^n/(n+12), abs_ratio=1/4*(n+12)/(n+13), limit=1/4, 5/8, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
13. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=13,B=13, rational ratio r=3/9. Set A=13>0,r=3/9∈(0,1), shift=13, alternating=false. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(13, 1/3, 13, No, a_n=13*(1/3)^n/(n+13), abs_ratio=1/3*(n+13)/(n+14), limit=1/3, 2/3, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (13, 1/3, 13, No, a_n=13*(1/3)^n/(n+13), abs_ratio=1/3*(n+13)/(n+14), limit=1/3, 2/3, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
14. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=14,B=14, rational ratio r=4/10. Set A=14>0,r=4/10∈(0,1), shift=14, alternating=true. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(14, 2/5, 14, Yes, a_n=14*(-1)^n*(2/5)^n/(n+14), abs_ratio=2/5*(n+14)/(n+15), limit=2/5, 7/10, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (14, 2/5, 14, Yes, a_n=14*(-1)^n*(2/5)^n/(n+14), abs_ratio=2/5*(n+14)/(n+15), limit=2/5, 7/10, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
15. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=15,B=15, rational ratio r=5/11. Set A=15>0,r=5/11∈(0,1), shift=15, alternating=false. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(15, 5/11, 15, No, a_n=15*(5/11)^n/(n+15), abs_ratio=5/11*(n+15)/(n+16), limit=5/11, 8/11, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (15, 5/11, 15, No, a_n=15*(5/11)^n/(n+15), abs_ratio=5/11*(n+15)/(n+16), limit=5/11, 8/11, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
16. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=16,B=16, rational ratio r=1/7. Set A=16>0,r=1/7∈(0,1), shift=16, alternating=true. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(16, 1/7, 16, Yes, a_n=16*(-1)^n*(1/7)^n/(n+16), abs_ratio=1/7*(n+16)/(n+17), limit=1/7, 4/7, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (16, 1/7, 16, Yes, a_n=16*(-1)^n*(1/7)^n/(n+16), abs_ratio=1/7*(n+16)/(n+17), limit=1/7, 4/7, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
17. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=17,B=17, rational ratio r=2/8. Set A=17>0,r=2/8∈(0,1), shift=17, alternating=false. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(17, 1/4, 17, No, a_n=17*(1/4)^n/(n+17), abs_ratio=1/4*(n+17)/(n+18), limit=1/4, 5/8, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (17, 1/4, 17, No, a_n=17*(1/4)^n/(n+17), abs_ratio=1/4*(n+17)/(n+18), limit=1/4, 5/8, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
18. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=18,B=18, rational ratio r=3/9. Set A=18>0,r=3/9∈(0,1), shift=18, alternating=true. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(18, 1/3, 18, Yes, a_n=18*(-1)^n*(1/3)^n/(n+18), abs_ratio=1/3*(n+18)/(n+19), limit=1/3, 2/3, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (18, 1/3, 18, Yes, a_n=18*(-1)^n*(1/3)^n/(n+18), abs_ratio=1/3*(n+18)/(n+19), limit=1/3, 2/3, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
19. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=19,B=19, rational ratio r=4/10. Set A=19>0,r=4/10∈(0,1), shift=19, alternating=false. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(19, 2/5, 19, No, a_n=19*(2/5)^n/(n+19), abs_ratio=2/5*(n+19)/(n+20), limit=2/5, 7/10, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (19, 2/5, 19, No, a_n=19*(2/5)^n/(n+19), abs_ratio=2/5*(n+19)/(n+20), limit=2/5, 7/10, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
20. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=20,B=20, rational ratio r=5/11. Set A=20>0,r=5/11∈(0,1), shift=20, alternating=true. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(20, 5/11, 20, Yes, a_n=20*(-1)^n*(5/11)^n/(n+20), abs_ratio=5/11*(n+20)/(n+21), limit=5/11, 8/11, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (20, 5/11, 20, Yes, a_n=20*(-1)^n*(5/11)^n/(n+20), abs_ratio=5/11*(n+20)/(n+21), limit=5/11, 8/11, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
21. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=21,B=21, rational ratio r=1/7. Set A=21>0,r=1/7∈(0,1), shift=21, alternating=false. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(21, 1/7, 21, No, a_n=21*(1/7)^n/(n+21), abs_ratio=1/7*(n+21)/(n+22), limit=1/7, 4/7, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (21, 1/7, 21, No, a_n=21*(1/7)^n/(n+21), abs_ratio=1/7*(n+21)/(n+22), limit=1/7, 4/7, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
22. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=22,B=22, rational ratio r=2/8. Set A=22>0,r=2/8∈(0,1), shift=22, alternating=true. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(22, 1/4, 22, Yes, a_n=22*(-1)^n*(1/4)^n/(n+22), abs_ratio=1/4*(n+22)/(n+23), limit=1/4, 5/8, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (22, 1/4, 22, Yes, a_n=22*(-1)^n*(1/4)^n/(n+22), abs_ratio=1/4*(n+22)/(n+23), limit=1/4, 5/8, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
23. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=23,B=23, rational ratio r=3/9. Set A=23>0,r=3/9∈(0,1), shift=23, alternating=false. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(23, 1/3, 23, No, a_n=23*(1/3)^n/(n+23), abs_ratio=1/3*(n+23)/(n+24), limit=1/3, 2/3, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (23, 1/3, 23, No, a_n=23*(1/3)^n/(n+23), abs_ratio=1/3*(n+23)/(n+24), limit=1/3, 2/3, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
24. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=24,B=24, rational ratio r=4/10. Set A=24>0,r=4/10∈(0,1), shift=24, alternating=true. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(24, 2/5, 24, Yes, a_n=24*(-1)^n*(2/5)^n/(n+24), abs_ratio=2/5*(n+24)/(n+25), limit=2/5, 7/10, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (24, 2/5, 24, Yes, a_n=24*(-1)^n*(2/5)^n/(n+24), abs_ratio=2/5*(n+24)/(n+25), limit=2/5, 7/10, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
25. A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit
Use exact scales A=25,B=25, rational ratio r=5/11. Set A=25>0,r=5/11∈(0,1), shift=25, alternating=false. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).
ratio is the whole rational r: the current additive answer writes A*(r)^n/(n+shift) with the complete base parenthesized, not p/(q^n). Original unparenthesized displays remain in the retained historical batch.
View answer and explanation
(25, 5/11, 25, No, a_n=25*(5/11)^n/(n+25), abs_ratio=5/11*(n+25)/(n+26), limit=5/11, 8/11, Yes, eventual_geometric_domination, absolutely_convergent)
The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (25, 5/11, 25, No, a_n=25*(5/11)^n/(n+25), abs_ratio=5/11*(n+25)/(n+26), limit=5/11, 8/11, Yes, eventual_geometric_domination, absolutely_convergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-nonconstant-convergent · candidate
26. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=1, exact ratio=6/3>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(1, 2, No, 2, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (1, 2, No, 2, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
27. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=2, exact ratio=7/4>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(2, 7/4, Yes, 7/4, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (2, 7/4, Yes, 7/4, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
28. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=3, exact ratio=8/5>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(3, 8/5, No, 8/5, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (3, 8/5, No, 8/5, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
29. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=4, exact ratio=9/6>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(4, 3/2, Yes, 3/2, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (4, 3/2, Yes, 3/2, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
30. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=5, exact ratio=10/7>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(5, 10/7, No, 10/7, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (5, 10/7, No, 10/7, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
31. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=6, exact ratio=6/3>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(6, 2, Yes, 2, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (6, 2, Yes, 2, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
32. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=7, exact ratio=7/4>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(7, 7/4, No, 7/4, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (7, 7/4, No, 7/4, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
33. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=8, exact ratio=8/5>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(8, 8/5, Yes, 8/5, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (8, 8/5, Yes, 8/5, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
34. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=9, exact ratio=9/6>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(9, 3/2, No, 3/2, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (9, 3/2, No, 3/2, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
35. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=10, exact ratio=10/7>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(10, 10/7, Yes, 10/7, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (10, 10/7, Yes, 10/7, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
36. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=11, exact ratio=6/3>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(11, 2, No, 2, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (11, 2, No, 2, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
37. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=12, exact ratio=7/4>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(12, 7/4, Yes, 7/4, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (12, 7/4, Yes, 7/4, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
38. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=13, exact ratio=8/5>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(13, 8/5, No, 8/5, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (13, 8/5, No, 8/5, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
39. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=14, exact ratio=9/6>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(14, 3/2, Yes, 3/2, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (14, 3/2, Yes, 3/2, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
40. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=15, exact ratio=10/7>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(15, 10/7, No, 10/7, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (15, 10/7, No, 10/7, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
41. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=16, exact ratio=6/3>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(16, 2, Yes, 2, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (16, 2, Yes, 2, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
42. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=17, exact ratio=7/4>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(17, 7/4, No, 7/4, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (17, 7/4, No, 7/4, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
43. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=18, exact ratio=8/5>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(18, 8/5, Yes, 8/5, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (18, 8/5, Yes, 8/5, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
44. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=19, exact ratio=9/6>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(19, 3/2, No, 3/2, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (19, 3/2, No, 3/2, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
45. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=20, exact ratio=10/7>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(20, 10/7, Yes, 10/7, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (20, 10/7, Yes, 10/7, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
46. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=21, exact ratio=6/3>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(21, 2, No, 2, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (21, 2, No, 2, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
47. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=22, exact ratio=7/4>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(22, 7/4, Yes, 7/4, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (22, 7/4, Yes, 7/4, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
48. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=23, exact ratio=8/5>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(23, 8/5, No, 8/5, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (23, 8/5, No, 8/5, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
49. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=24, exact ratio=9/6>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(24, 3/2, Yes, 3/2, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (24, 3/2, Yes, 3/2, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
50. An Absolute Geometric Ratio above One Violates the Necessary Term Limit
Use positive scale=25, exact ratio=10/7>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).
View answer and explanation
(25, 10/7, No, 10/7, terms_do_not_tend_to_zero, divergent)
The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (25, 10/7, No, 10/7, terms_do_not_tend_to_zero, divergent).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-geometric-divergent · candidate
51. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
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).
View answer and explanation
(1, 1, abs_ratio=((n+1)/(n+2))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (1, 1, abs_ratio=((n+1)/(n+2))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
52. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=2∈{1,2}, shift=2≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(2, 2, abs_ratio=((n+2)/(n+3))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (2, 2, abs_ratio=((n+2)/(n+3))^2, 1, convergent_p_series, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
53. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=1∈{1,2}, shift=3≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(1, 3, abs_ratio=((n+3)/(n+4))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (1, 3, abs_ratio=((n+3)/(n+4))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
54. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
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).
View answer and explanation
(2, 4, abs_ratio=((n+4)/(n+5))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (2, 4, abs_ratio=((n+4)/(n+5))^2, 1, convergent_p_series, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
55. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=1∈{1,2}, shift=5≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(1, 5, abs_ratio=((n+5)/(n+6))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (1, 5, abs_ratio=((n+5)/(n+6))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
56. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=2∈{1,2}, shift=6≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(2, 6, abs_ratio=((n+6)/(n+7))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (2, 6, abs_ratio=((n+6)/(n+7))^2, 1, convergent_p_series, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
57. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
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).
View answer and explanation
(1, 7, abs_ratio=((n+7)/(n+8))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (1, 7, abs_ratio=((n+7)/(n+8))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
58. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=2∈{1,2}, shift=8≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(2, 8, abs_ratio=((n+8)/(n+9))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (2, 8, abs_ratio=((n+8)/(n+9))^2, 1, convergent_p_series, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
59. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=1∈{1,2}, shift=9≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(1, 9, abs_ratio=((n+9)/(n+10))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (1, 9, abs_ratio=((n+9)/(n+10))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
60. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=2∈{1,2}, shift=10≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(2, 10, abs_ratio=((n+10)/(n+11))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (2, 10, abs_ratio=((n+10)/(n+11))^2, 1, convergent_p_series, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
61. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=1∈{1,2}, shift=11≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(1, 11, abs_ratio=((n+11)/(n+12))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (1, 11, abs_ratio=((n+11)/(n+12))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
62. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=2∈{1,2}, shift=12≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(2, 12, abs_ratio=((n+12)/(n+13))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (2, 12, abs_ratio=((n+12)/(n+13))^2, 1, convergent_p_series, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
63. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=1∈{1,2}, shift=13≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(1, 13, abs_ratio=((n+13)/(n+14))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (1, 13, abs_ratio=((n+13)/(n+14))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
64. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=2∈{1,2}, shift=14≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(2, 14, abs_ratio=((n+14)/(n+15))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (2, 14, abs_ratio=((n+14)/(n+15))^2, 1, convergent_p_series, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
65. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=1∈{1,2}, shift=15≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(1, 15, abs_ratio=((n+15)/(n+16))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (1, 15, abs_ratio=((n+15)/(n+16))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
66. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=2∈{1,2}, shift=16≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(2, 16, abs_ratio=((n+16)/(n+17))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (2, 16, abs_ratio=((n+16)/(n+17))^2, 1, convergent_p_series, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
67. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=1∈{1,2}, shift=17≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(1, 17, abs_ratio=((n+17)/(n+18))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (1, 17, abs_ratio=((n+17)/(n+18))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
68. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=2∈{1,2}, shift=18≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(2, 18, abs_ratio=((n+18)/(n+19))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (2, 18, abs_ratio=((n+18)/(n+19))^2, 1, convergent_p_series, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
69. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=1∈{1,2}, shift=19≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(1, 19, abs_ratio=((n+19)/(n+20))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (1, 19, abs_ratio=((n+19)/(n+20))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
70. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=2∈{1,2}, shift=20≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(2, 20, abs_ratio=((n+20)/(n+21))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (2, 20, abs_ratio=((n+20)/(n+21))^2, 1, convergent_p_series, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
71. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=1∈{1,2}, shift=21≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(1, 21, abs_ratio=((n+21)/(n+22))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (1, 21, abs_ratio=((n+21)/(n+22))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
72. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=2∈{1,2}, shift=22≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(2, 22, abs_ratio=((n+22)/(n+23))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (2, 22, abs_ratio=((n+22)/(n+23))^2, 1, convergent_p_series, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
73. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=1∈{1,2}, shift=23≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(1, 23, abs_ratio=((n+23)/(n+24))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (1, 23, abs_ratio=((n+23)/(n+24))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
74. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
Use a_n=1/(n+shift)^p with p=2∈{1,2}, shift=24≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).
View answer and explanation
(2, 24, abs_ratio=((n+24)/(n+25))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (2, 24, abs_ratio=((n+24)/(n+25))^2, 1, convergent_p_series, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
75. Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence
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).
View answer and explanation
(1, 25, abs_ratio=((n+25)/(n+26))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)
The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (1, 25, abs_ratio=((n+25)/(n+26))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-limit-one-inconclusive · candidate
76. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=zero_terms and positive integer scale=2. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(zero_terms, 2, a_n=2_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (zero_terms, 2, a_n=2_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
77. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=finite_samples and positive integer scale=3. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(finite_samples, 3, ratios_equal_1/2_for_n<=3_then_equal_2, finite_prefix_cannot_determine_ratio_limit, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (finite_samples, 3, ratios_equal_1/2_for_n<=3_then_equal_2, finite_prefix_cannot_determine_ratio_limit, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
78. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=missing_limit and positive integer scale=4. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(missing_limit, 4, absolute_ratios_alternate_between_1/5_and_5, limit_does_not_exist, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (missing_limit, 4, absolute_ratios_alternate_between_1/5_and_5, limit_does_not_exist, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
79. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=signed_without_absolute and positive integer scale=5. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(signed_without_absolute, 5, a_n=(-1)^n*5*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (signed_without_absolute, 5, a_n=(-1)^n*5*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
80. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=zero_terms and positive integer scale=6. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(zero_terms, 6, a_n=6_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (zero_terms, 6, a_n=6_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
81. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=finite_samples and positive integer scale=7. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(finite_samples, 7, ratios_equal_1/2_for_n<=7_then_equal_2, finite_prefix_cannot_determine_ratio_limit, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (finite_samples, 7, ratios_equal_1/2_for_n<=7_then_equal_2, finite_prefix_cannot_determine_ratio_limit, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
82. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=missing_limit and positive integer scale=8. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(missing_limit, 8, absolute_ratios_alternate_between_1/9_and_9, limit_does_not_exist, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (missing_limit, 8, absolute_ratios_alternate_between_1/9_and_9, limit_does_not_exist, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
83. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=signed_without_absolute and positive integer scale=9. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(signed_without_absolute, 9, a_n=(-1)^n*9*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (signed_without_absolute, 9, a_n=(-1)^n*9*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
84. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=zero_terms and positive integer scale=10. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(zero_terms, 10, a_n=10_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (zero_terms, 10, a_n=10_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
85. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=finite_samples and positive integer scale=11. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(finite_samples, 11, ratios_equal_1/2_for_n<=11_then_equal_2, finite_prefix_cannot_determine_ratio_limit, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (finite_samples, 11, ratios_equal_1/2_for_n<=11_then_equal_2, finite_prefix_cannot_determine_ratio_limit, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
86. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=missing_limit and positive integer scale=12. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(missing_limit, 12, absolute_ratios_alternate_between_1/13_and_13, limit_does_not_exist, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (missing_limit, 12, absolute_ratios_alternate_between_1/13_and_13, limit_does_not_exist, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
87. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=signed_without_absolute and positive integer scale=13. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(signed_without_absolute, 13, a_n=(-1)^n*13*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (signed_without_absolute, 13, a_n=(-1)^n*13*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
88. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=zero_terms and positive integer scale=14. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(zero_terms, 14, a_n=14_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (zero_terms, 14, a_n=14_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
89. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=finite_samples and positive integer scale=15. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(finite_samples, 15, ratios_equal_1/2_for_n<=15_then_equal_2, finite_prefix_cannot_determine_ratio_limit, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (finite_samples, 15, ratios_equal_1/2_for_n<=15_then_equal_2, finite_prefix_cannot_determine_ratio_limit, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
90. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=missing_limit and positive integer scale=16. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(missing_limit, 16, absolute_ratios_alternate_between_1/17_and_17, limit_does_not_exist, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (missing_limit, 16, absolute_ratios_alternate_between_1/17_and_17, limit_does_not_exist, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
91. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=signed_without_absolute and positive integer scale=17. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(signed_without_absolute, 17, a_n=(-1)^n*17*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (signed_without_absolute, 17, a_n=(-1)^n*17*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
92. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=zero_terms and positive integer scale=18. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(zero_terms, 18, a_n=18_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (zero_terms, 18, a_n=18_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
93. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=finite_samples and positive integer scale=19. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(finite_samples, 19, ratios_equal_1/2_for_n<=19_then_equal_2, finite_prefix_cannot_determine_ratio_limit, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (finite_samples, 19, ratios_equal_1/2_for_n<=19_then_equal_2, finite_prefix_cannot_determine_ratio_limit, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
94. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=missing_limit and positive integer scale=20. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(missing_limit, 20, absolute_ratios_alternate_between_1/21_and_21, limit_does_not_exist, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (missing_limit, 20, absolute_ratios_alternate_between_1/21_and_21, limit_does_not_exist, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
95. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=signed_without_absolute and positive integer scale=21. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(signed_without_absolute, 21, a_n=(-1)^n*21*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (signed_without_absolute, 21, a_n=(-1)^n*21*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
96. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=zero_terms and positive integer scale=22. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(zero_terms, 22, a_n=22_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (zero_terms, 22, a_n=22_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
97. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=finite_samples and positive integer scale=23. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(finite_samples, 23, ratios_equal_1/2_for_n<=23_then_equal_2, finite_prefix_cannot_determine_ratio_limit, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (finite_samples, 23, ratios_equal_1/2_for_n<=23_then_equal_2, finite_prefix_cannot_determine_ratio_limit, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
98. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=missing_limit and positive integer scale=24. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(missing_limit, 24, absolute_ratios_alternate_between_1/25_and_25, limit_does_not_exist, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (missing_limit, 24, absolute_ratios_alternate_between_1/25_and_25, limit_does_not_exist, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
99. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=signed_without_absolute and positive integer scale=25. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(signed_without_absolute, 25, a_n=(-1)^n*25*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (signed_without_absolute, 25, a_n=(-1)^n*25*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
100. Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises
Use mode=zero_terms and positive integer scale=26. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).
The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1.
View answer and explanation
(zero_terms, 26, a_n=26_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No)
Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (zero_terms, 26, a_n=26_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No).
Reviewed step-by-step derivation
Ratio test for series · qf-mod48-ratio-test-boundary · candidate
Canonical question JSONL · English JSONL · Validation report