{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0001","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e004032ad8cf707a305398db0075f4eb0fdcab0e89f30654cf01983f6af1a19f","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172001,"semantic_template":"ratio-nonconstant-convergent|{alternating:false,ratio:[1,7],scale:[1,1],shift:1}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0002","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9aa0311fdd28a5c89ec9674c331dee597cc8d5496f014020109bcf5aa338c03","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172002,"semantic_template":"ratio-nonconstant-convergent|{alternating:true,ratio:[1,4],scale:[2,1],shift:2}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0003","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1378340f006f4ae94c7ef0435ef40778f3f9b989dca2e535028f30076fc0e9e3","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172003,"semantic_template":"ratio-nonconstant-convergent|{alternating:false,ratio:[1,3],scale:[3,1],shift:3}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0004","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59782aefec449e143f47b7f3403fee7eb47878f3d387f18ca997553eb851e922","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172004,"semantic_template":"ratio-nonconstant-convergent|{alternating:true,ratio:[2,5],scale:[4,1],shift:4}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0005","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3451c5e396c41ff8b08b32da8a0f8437fbe7207b646fd05892177879444150bf","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172005,"semantic_template":"ratio-nonconstant-convergent|{alternating:false,ratio:[5,11],scale:[5,1],shift:5}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0006","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3554a167c663728e0916aeee66aef23c674e607b0322bdae795e1e9b54af00ef","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172006,"semantic_template":"ratio-nonconstant-convergent|{alternating:true,ratio:[1,7],scale:[6,1],shift:6}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0007","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"762773f96ec120087e9d636226f104d257f711b2cd0816b8f128d83ba3c181c5","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172007,"semantic_template":"ratio-nonconstant-convergent|{alternating:false,ratio:[1,4],scale:[7,1],shift:7}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0008","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8e131cb9e07ad048468b287a675be4444ce634b3c5bca29ad7cf3c9f98868d3","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172008,"semantic_template":"ratio-nonconstant-convergent|{alternating:true,ratio:[1,3],scale:[8,1],shift:8}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0009","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce6343628c923eba88e33e1ba6a8f10f6e08868e81682872a3fc1a78ad1d2885","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172009,"semantic_template":"ratio-nonconstant-convergent|{alternating:false,ratio:[2,5],scale:[9,1],shift:9}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0010","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b3f90c11a59f5691ed8b4fa77cac3641e6243ba8452a7ffe3f39a1d44a357fa","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172010,"semantic_template":"ratio-nonconstant-convergent|{alternating:true,ratio:[5,11],scale:[10,1],shift:10}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0011","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff3e3aacf7f80fdb7a91d647130b9ca4d09f3a30ba13db51abdd9364f8c6bcbe","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172011,"semantic_template":"ratio-nonconstant-convergent|{alternating:false,ratio:[1,7],scale:[11,1],shift:11}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0012","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a7ed4c3c73779716ea332d76b65bbea8b9f12472b242f3c0a50c5e24c53d986","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172012,"semantic_template":"ratio-nonconstant-convergent|{alternating:true,ratio:[1,4],scale:[12,1],shift:12}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0013","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a36d6b2350c5870faf7f716261de056cacb1e4b0245853dd75a0af0f99660e2","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172013,"semantic_template":"ratio-nonconstant-convergent|{alternating:false,ratio:[1,3],scale:[13,1],shift:13}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0014","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c288fb94de33215f9d2a56c6c639930356c2abfde4191f82c497eb46272e7bc","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172014,"semantic_template":"ratio-nonconstant-convergent|{alternating:true,ratio:[2,5],scale:[14,1],shift:14}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0015","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc9553af7a3ba7a470a2d7acb11ccac08270823d08ba837ac8b8018eba65dd8d","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172015,"semantic_template":"ratio-nonconstant-convergent|{alternating:false,ratio:[5,11],scale:[15,1],shift:15}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0016","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48ad3640773f724c7eecfe0aea5d818268ad6b1fe77835b9edef3ae0fcc43d20","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172016,"semantic_template":"ratio-nonconstant-convergent|{alternating:true,ratio:[1,7],scale:[16,1],shift:16}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0017","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e20b51b564b7b1b7d4e94ff233a3024b02ebb3be2d87336fe02d4bd9078be639","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172017,"semantic_template":"ratio-nonconstant-convergent|{alternating:false,ratio:[1,4],scale:[17,1],shift:17}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0018","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67f2f5ad644dddb96a17dd3b630a13ea2200cf75b4271c3c2a09bc3bde992fca","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172018,"semantic_template":"ratio-nonconstant-convergent|{alternating:true,ratio:[1,3],scale:[18,1],shift:18}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0019","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f22aaa2d63cab570dd9396543d80c782bf205363ea1e6904be05e901dd20c561","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172019,"semantic_template":"ratio-nonconstant-convergent|{alternating:false,ratio:[2,5],scale:[19,1],shift:19}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0020","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"836ee0d37df3e69075d51c6cb813d487448f332074bda1816eaa94c6f8a13ccd","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172020,"semantic_template":"ratio-nonconstant-convergent|{alternating:true,ratio:[5,11],scale:[20,1],shift:20}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0021","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7c82ad0948c1980e40c811e68990b9a2d00ed3ae12cf1db6d47ca985f9ec970","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172021,"semantic_template":"ratio-nonconstant-convergent|{alternating:false,ratio:[1,7],scale:[21,1],shift:21}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0022","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75fb7b4d14f23ae120d657e66ab79798469f73998ed91ac0f768d21c227de7c5","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172022,"semantic_template":"ratio-nonconstant-convergent|{alternating:true,ratio:[1,4],scale:[22,1],shift:22}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0023","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"671f871f2ec8e3fd096d1ecc1cb3e9be662eabbbd0a34020f6729cbcc2591e05","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172023,"semantic_template":"ratio-nonconstant-convergent|{alternating:false,ratio:[1,3],scale:[23,1],shift:23}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0024","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f6cd2abd99ed9dfe25339338def85ed913aef08bf86f195a2b30aafaab0eb4e","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172024,"semantic_template":"ratio-nonconstant-convergent|{alternating:true,ratio:[2,5],scale:[24,1],shift:24}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0025","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"099de94dd1db2f3ca6f988f3896bc2ade5b4febaa23f7ff88978e1b6b88d5869","family_id":"qf-mod48-ratio-nonconstant-convergent","seed":202611172025,"semantic_template":"ratio-nonconstant-convergent|{alternating:false,ratio:[5,11],scale:[25,1],shift:25}"},"fields":{"title":"A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit","stem":"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).","answer":{"kind":"expression","value":"(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)","explanation":"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)."},"skills":["A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0026","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9572b9431fa8637e3d386d155ec9bee968baddf274452f1a300c5944a65e10aa","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172026,"semantic_template":"ratio-geometric-divergent|{alternating:false,ratio:[2,1],scale:[1,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=1, exact ratio=6/3>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(1, 2, No, 2, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0027","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3601e0b21a9602d768b996ce6efe19c9ae8b0c92d6ec738804e729ab9af8323f","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172027,"semantic_template":"ratio-geometric-divergent|{alternating:true,ratio:[7,4],scale:[2,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=2, exact ratio=7/4>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(2, 7/4, Yes, 7/4, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0028","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53569c5bf5d75d4b2a85e60836fd07f94fa24d0f503127768c764ccd6290b731","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172028,"semantic_template":"ratio-geometric-divergent|{alternating:false,ratio:[8,5],scale:[3,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=3, exact ratio=8/5>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(3, 8/5, No, 8/5, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0029","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f45a1c51ea82cad40b1c00cc1c8fa5fe0b8e12e1135a54f29c7b056046c0dfe9","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172029,"semantic_template":"ratio-geometric-divergent|{alternating:true,ratio:[3,2],scale:[4,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=4, exact ratio=9/6>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(4, 3/2, Yes, 3/2, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0030","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c670282d2403104b3beb03c9f913c306a38712f0446118da4ab004803b739190","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172030,"semantic_template":"ratio-geometric-divergent|{alternating:false,ratio:[10,7],scale:[5,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=5, exact ratio=10/7>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(5, 10/7, No, 10/7, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0031","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6adbf7f287aba7a616d23e9e1ba1e0824e447b8fc05b406853150dc3cc193a14","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172031,"semantic_template":"ratio-geometric-divergent|{alternating:true,ratio:[2,1],scale:[6,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=6, exact ratio=6/3>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(6, 2, Yes, 2, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0032","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"104ec2938df9e1834452588a99dd2a1229087209dd6bd8d4a0da35014fb7d211","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172032,"semantic_template":"ratio-geometric-divergent|{alternating:false,ratio:[7,4],scale:[7,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=7, exact ratio=7/4>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(7, 7/4, No, 7/4, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0033","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a30217db2946af51c6b0b79a15ccdf10ec7fd440685a7cb116ba3f4a10d7331","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172033,"semantic_template":"ratio-geometric-divergent|{alternating:true,ratio:[8,5],scale:[8,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=8, exact ratio=8/5>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(8, 8/5, Yes, 8/5, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0034","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5f53bd1d4a21d5f679ebff29847bd5096b00be498fb1164ca68c54a37b808cb","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172034,"semantic_template":"ratio-geometric-divergent|{alternating:false,ratio:[3,2],scale:[9,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=9, exact ratio=9/6>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(9, 3/2, No, 3/2, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0035","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd24448259108b8b958ac63c8eb31e5d7192426558649566775d68d92de8cd42","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172035,"semantic_template":"ratio-geometric-divergent|{alternating:true,ratio:[10,7],scale:[10,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=10, exact ratio=10/7>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(10, 10/7, Yes, 10/7, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0036","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03dab2b179893b5c614f3c44552186c5593096f2a634e0d48733acc7c62ee25f","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172036,"semantic_template":"ratio-geometric-divergent|{alternating:false,ratio:[2,1],scale:[11,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=11, exact ratio=6/3>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(11, 2, No, 2, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0037","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3574646ace5e8a10f41643d53597a6f3c588831d04eb6df4024c237906d5121d","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172037,"semantic_template":"ratio-geometric-divergent|{alternating:true,ratio:[7,4],scale:[12,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=12, exact ratio=7/4>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(12, 7/4, Yes, 7/4, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0038","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8c9be462164df07a7ae73e30932ac2597ab07d6cae1e9dd80ae58214f1eed1a","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172038,"semantic_template":"ratio-geometric-divergent|{alternating:false,ratio:[8,5],scale:[13,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=13, exact ratio=8/5>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(13, 8/5, No, 8/5, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0039","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"261bd4472ead6b83bc40c9971d8c8cbdfefd7664e178e5ee3a1d6f5870b392b8","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172039,"semantic_template":"ratio-geometric-divergent|{alternating:true,ratio:[3,2],scale:[14,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=14, exact ratio=9/6>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(14, 3/2, Yes, 3/2, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0040","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bac36b954a75a6d9f23221349f76241a1fb3f89e607b021eac0e94a05d05f5a9","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172040,"semantic_template":"ratio-geometric-divergent|{alternating:false,ratio:[10,7],scale:[15,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=15, exact ratio=10/7>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(15, 10/7, No, 10/7, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0041","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e2a2e64eff0c9eecff8f0e78da38633c931e105c4932019a88262c2dacbcefa","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172041,"semantic_template":"ratio-geometric-divergent|{alternating:true,ratio:[2,1],scale:[16,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=16, exact ratio=6/3>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(16, 2, Yes, 2, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0042","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30e83c7580c65b4fbac2db86639699a5b15a100691215e703b2986d8348b392f","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172042,"semantic_template":"ratio-geometric-divergent|{alternating:false,ratio:[7,4],scale:[17,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=17, exact ratio=7/4>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(17, 7/4, No, 7/4, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0043","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"303ce956d683b455bbe00707d29eb27b03ca17ab244199d5cad44b3b40430a13","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172043,"semantic_template":"ratio-geometric-divergent|{alternating:true,ratio:[8,5],scale:[18,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=18, exact ratio=8/5>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(18, 8/5, Yes, 8/5, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0044","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1b062737611de41e2b0eb1681c94d6076455e89ceac8e9ed4255bb7657a4a74","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172044,"semantic_template":"ratio-geometric-divergent|{alternating:false,ratio:[3,2],scale:[19,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=19, exact ratio=9/6>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(19, 3/2, No, 3/2, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0045","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cf5eeafab6bf53be8a7a6dc0c7962a63946fab5847d11d23213a70254ff8029","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172045,"semantic_template":"ratio-geometric-divergent|{alternating:true,ratio:[10,7],scale:[20,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=20, exact ratio=10/7>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(20, 10/7, Yes, 10/7, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0046","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d29ff82fd9d916f826c54ce44a4db0b4332ce01eea74567b0a7e58ccd01cfb54","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172046,"semantic_template":"ratio-geometric-divergent|{alternating:false,ratio:[2,1],scale:[21,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=21, exact ratio=6/3>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(21, 2, No, 2, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0047","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d0fafe7dbfcc1707107fa5027aa0de1353dcf0abf65cf9e6163f4a2ccc1f39f","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172047,"semantic_template":"ratio-geometric-divergent|{alternating:true,ratio:[7,4],scale:[22,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=22, exact ratio=7/4>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(22, 7/4, Yes, 7/4, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0048","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"788b3384be0b95551e6294dae3151a9f1e90a2491ac356f230323174ce4083d1","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172048,"semantic_template":"ratio-geometric-divergent|{alternating:false,ratio:[8,5],scale:[23,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=23, exact ratio=8/5>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(23, 8/5, No, 8/5, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0049","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1360994715f5d43c6862c4167c31e627ea828fcc8c0de0e844b1b86224262839","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172049,"semantic_template":"ratio-geometric-divergent|{alternating:true,ratio:[3,2],scale:[24,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=24, exact ratio=9/6>1 and alternating=true. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(24, 3/2, Yes, 3/2, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0050","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30a18fff673bd643e600db9ca8cd7ff49948969286324a783b6396ad3b91e60d","family_id":"qf-mod48-ratio-geometric-divergent","seed":202611172050,"semantic_template":"ratio-geometric-divergent|{alternating:false,ratio:[10,7],scale:[25,1]}"},"fields":{"title":"An Absolute Geometric Ratio above One Violates the Necessary Term Limit","stem":"Use positive scale=25, exact ratio=10/7>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).","answer":{"kind":"expression","value":"(25, 10/7, No, 10/7, terms_do_not_tend_to_zero, divergent)","explanation":"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)."},"skills":["An Absolute Geometric Ratio above One Violates the Necessary Term Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0051","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e14bc1837d0454579ae680029edeb80f435071b1c108419181e00a0c17e6244f","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172051,"semantic_template":"ratio-limit-one-inconclusive|{exponent:1,shift:1}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(1, 1, abs_ratio=((n+1)/(n+2))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0052","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4185169b6dfd1d72216096ad98cba0b70d753e422aa54a1d51e1471c313cfb1","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172052,"semantic_template":"ratio-limit-one-inconclusive|{exponent:2,shift:2}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(2, 2, abs_ratio=((n+2)/(n+3))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0053","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"493f63c26c043d1229f8779f3c4a60e424da3f267881f2837153bdcc8a7244ca","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172053,"semantic_template":"ratio-limit-one-inconclusive|{exponent:1,shift:3}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(1, 3, abs_ratio=((n+3)/(n+4))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0054","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84e7ff9e6a48e3d5d58c680eec55722d0e21d419a8d670f4774d1ca2bd181326","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172054,"semantic_template":"ratio-limit-one-inconclusive|{exponent:2,shift:4}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(2, 4, abs_ratio=((n+4)/(n+5))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0055","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec50ac2e40654e69b4930e6f8a06dfcbb7164238735b4954b38cb6b8dd670a40","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172055,"semantic_template":"ratio-limit-one-inconclusive|{exponent:1,shift:5}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(1, 5, abs_ratio=((n+5)/(n+6))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0056","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f46d9ea69b350732e0a8a1d639dd9e59bcfdbcf170e3ab37055120effc140682","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172056,"semantic_template":"ratio-limit-one-inconclusive|{exponent:2,shift:6}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(2, 6, abs_ratio=((n+6)/(n+7))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0057","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"439e0a9929c9c45b4e5963957b8d3ab90cc04ca2da913d0584afd7a574483b4e","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172057,"semantic_template":"ratio-limit-one-inconclusive|{exponent:1,shift:7}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(1, 7, abs_ratio=((n+7)/(n+8))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0058","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd1d52038f007ae70892e5c63c77c858ae3e1bab03aaaba1b73302c1726229cf","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172058,"semantic_template":"ratio-limit-one-inconclusive|{exponent:2,shift:8}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(2, 8, abs_ratio=((n+8)/(n+9))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0059","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32cc25abc277013465e84e5e2a0af6dc0a68bb1dfca3a055e14eb31c9666fa38","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172059,"semantic_template":"ratio-limit-one-inconclusive|{exponent:1,shift:9}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(1, 9, abs_ratio=((n+9)/(n+10))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0060","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de3e9fe6924d73d563d8f5dcffdabe1a0b4ebb48d8c702f3cc26a920e9604380","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172060,"semantic_template":"ratio-limit-one-inconclusive|{exponent:2,shift:10}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(2, 10, abs_ratio=((n+10)/(n+11))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0061","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e2b62ddbb0e3466d8b34b22f6d7d0cd617f638526a821ae43b1173a3f6cc322","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172061,"semantic_template":"ratio-limit-one-inconclusive|{exponent:1,shift:11}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(1, 11, abs_ratio=((n+11)/(n+12))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0062","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9445db65b5a84dfe6b5a589028ff5fb0d580bd92196c35e3a49decd3447308c2","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172062,"semantic_template":"ratio-limit-one-inconclusive|{exponent:2,shift:12}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(2, 12, abs_ratio=((n+12)/(n+13))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0063","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f292161dc45ee8f403a435696b1fc09a72dfe3fb62da6212f6e30fa8081f086","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172063,"semantic_template":"ratio-limit-one-inconclusive|{exponent:1,shift:13}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(1, 13, abs_ratio=((n+13)/(n+14))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0064","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22c6c5619025235c0224e72012a8cb621421f7b237762eeacaf4ef3cd8f580b4","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172064,"semantic_template":"ratio-limit-one-inconclusive|{exponent:2,shift:14}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(2, 14, abs_ratio=((n+14)/(n+15))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0065","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6aacda35b3eabed7a33bf0ae79fd0a0fc7d07504f559ceef58e4f5cdaf60a326","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172065,"semantic_template":"ratio-limit-one-inconclusive|{exponent:1,shift:15}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(1, 15, abs_ratio=((n+15)/(n+16))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0066","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd0176070696c431bdf74497db1ed928516a6ccd7ab4432566979e69ab965e5c","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172066,"semantic_template":"ratio-limit-one-inconclusive|{exponent:2,shift:16}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(2, 16, abs_ratio=((n+16)/(n+17))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0067","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e40a99611ca23699e888cbec1d8f627ecc607f8e0171afa7191f7fc1c0291726","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172067,"semantic_template":"ratio-limit-one-inconclusive|{exponent:1,shift:17}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(1, 17, abs_ratio=((n+17)/(n+18))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0068","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c544e1bcb39c9da34bc4a3b7ed6f4686ef010edab25d1ed48b6702e1458d2bc","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172068,"semantic_template":"ratio-limit-one-inconclusive|{exponent:2,shift:18}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(2, 18, abs_ratio=((n+18)/(n+19))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0069","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4587ff4684010084e74baa2b47daf38109bb83cd1829c09cf2e79bd4894d9110","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172069,"semantic_template":"ratio-limit-one-inconclusive|{exponent:1,shift:19}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(1, 19, abs_ratio=((n+19)/(n+20))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0070","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9b4f5f2c3114952952c50e44c10c0eedc548bca25f527cc9eee49cf9ef667a0","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172070,"semantic_template":"ratio-limit-one-inconclusive|{exponent:2,shift:20}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(2, 20, abs_ratio=((n+20)/(n+21))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0071","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f4e3195995aa17acc1d8677b49ebfc4f44f2165fcb55586214c75eb87f72ee7","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172071,"semantic_template":"ratio-limit-one-inconclusive|{exponent:1,shift:21}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(1, 21, abs_ratio=((n+21)/(n+22))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0072","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0411f9287b590c501b82579a4c216520d80739043dbea8add872de50e4da2e5","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172072,"semantic_template":"ratio-limit-one-inconclusive|{exponent:2,shift:22}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(2, 22, abs_ratio=((n+22)/(n+23))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0073","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e627f4a7472b80c729c9664370102869080f66674e89399e0763224a7415faa","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172073,"semantic_template":"ratio-limit-one-inconclusive|{exponent:1,shift:23}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(1, 23, abs_ratio=((n+23)/(n+24))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0074","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee18239292cf5c00d2e3033940ef05a873081c36217508656287237633582adc","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172074,"semantic_template":"ratio-limit-one-inconclusive|{exponent:2,shift:24}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(2, 24, abs_ratio=((n+24)/(n+25))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0075","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf29d8883ba4b2b839c278df7db3a31c52361b86283205863badf933dca6b7d2","family_id":"qf-mod48-ratio-limit-one-inconclusive","seed":202611172075,"semantic_template":"ratio-limit-one-inconclusive|{exponent:1,shift:25}"},"fields":{"title":"Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence","stem":"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).","answer":{"kind":"expression","value":"(1, 25, abs_ratio=((n+25)/(n+26))^1, 1, divergent_harmonic_tail, L_equal_one_is_inconclusive)","explanation":"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)."},"skills":["Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0076","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0005d3a61db4e411e0fcb482492a90fe9cfa1b3dc4b7e296cf63a374d24397c9","family_id":"qf-mod48-ratio-test-boundary","seed":202611172076,"semantic_template":"ratio-test-boundary|{mode:\"zero_terms\",scale:2}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=zero_terms and positive integer scale=2. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(zero_terms, 2, a_n=2_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0077","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa2efc8d197f757b912415598719e66b5468e9fee8ba0d0a159dd05fc90711f9","family_id":"qf-mod48-ratio-test-boundary","seed":202611172077,"semantic_template":"ratio-test-boundary|{mode:\"finite_samples\",scale:3}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=finite_samples and positive integer scale=3. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(finite_samples, 3, ratios_equal_1/2_for_n<=3_then_equal_2, finite_prefix_cannot_determine_ratio_limit, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0078","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"606989a3c15749c13722a4f8f15845787ff89c963c270e32327d3a166b45facd","family_id":"qf-mod48-ratio-test-boundary","seed":202611172078,"semantic_template":"ratio-test-boundary|{mode:\"missing_limit\",scale:4}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=missing_limit and positive integer scale=4. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(missing_limit, 4, absolute_ratios_alternate_between_1/5_and_5, limit_does_not_exist, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0079","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35f072c527f4a9977c254f2f09d39aeddd4de2befaf4b39d12a8f5548c28324b","family_id":"qf-mod48-ratio-test-boundary","seed":202611172079,"semantic_template":"ratio-test-boundary|{mode:\"signed_without_absolute\",scale:5}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=signed_without_absolute and positive integer scale=5. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(signed_without_absolute, 5, a_n=(-1)^n*5*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0080","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5156f8dbb5ae7870285514feac1ff68334e43c2341f99725b7f85bf1ad9ee555","family_id":"qf-mod48-ratio-test-boundary","seed":202611172080,"semantic_template":"ratio-test-boundary|{mode:\"zero_terms\",scale:6}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=zero_terms and positive integer scale=6. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(zero_terms, 6, a_n=6_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0081","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be546b337271f27a7de4d3811f8607edab9af613c100875d17df20a9db47549e","family_id":"qf-mod48-ratio-test-boundary","seed":202611172081,"semantic_template":"ratio-test-boundary|{mode:\"finite_samples\",scale:7}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=finite_samples and positive integer scale=7. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(finite_samples, 7, ratios_equal_1/2_for_n<=7_then_equal_2, finite_prefix_cannot_determine_ratio_limit, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0082","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77537fc72e8743436303a51fd9e6fdc287d9c5475194afe88caab35a337277d3","family_id":"qf-mod48-ratio-test-boundary","seed":202611172082,"semantic_template":"ratio-test-boundary|{mode:\"missing_limit\",scale:8}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=missing_limit and positive integer scale=8. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(missing_limit, 8, absolute_ratios_alternate_between_1/9_and_9, limit_does_not_exist, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0083","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4acf9ae7628ec62ee77386975a708f475f8fbdc40496fb55ee5da8ddb194456","family_id":"qf-mod48-ratio-test-boundary","seed":202611172083,"semantic_template":"ratio-test-boundary|{mode:\"signed_without_absolute\",scale:9}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=signed_without_absolute and positive integer scale=9. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(signed_without_absolute, 9, a_n=(-1)^n*9*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0084","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c99b9588b50774bc99836895fd0710e9fdaa24ee04586ca9582ff737b1df1bd3","family_id":"qf-mod48-ratio-test-boundary","seed":202611172084,"semantic_template":"ratio-test-boundary|{mode:\"zero_terms\",scale:10}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=zero_terms and positive integer scale=10. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(zero_terms, 10, a_n=10_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0085","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aec6ed5f75119d385b5b3d70ede440b753ca64170dd608fa582891b8e6268c69","family_id":"qf-mod48-ratio-test-boundary","seed":202611172085,"semantic_template":"ratio-test-boundary|{mode:\"finite_samples\",scale:11}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=finite_samples and positive integer scale=11. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(finite_samples, 11, ratios_equal_1/2_for_n<=11_then_equal_2, finite_prefix_cannot_determine_ratio_limit, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0086","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20cb55fdfaecccb8872bb857a46e55c4e06df02e0354fd2ddfd0c4acd7f67762","family_id":"qf-mod48-ratio-test-boundary","seed":202611172086,"semantic_template":"ratio-test-boundary|{mode:\"missing_limit\",scale:12}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=missing_limit and positive integer scale=12. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(missing_limit, 12, absolute_ratios_alternate_between_1/13_and_13, limit_does_not_exist, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0087","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2a82664a810f88f213063336bc2f1f04283941bb79822443df81429e08c667a","family_id":"qf-mod48-ratio-test-boundary","seed":202611172087,"semantic_template":"ratio-test-boundary|{mode:\"signed_without_absolute\",scale:13}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=signed_without_absolute and positive integer scale=13. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(signed_without_absolute, 13, a_n=(-1)^n*13*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0088","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6d26785d75e8c762c652add613868def5009a08f1d34f515c9aa0cd55c43a78","family_id":"qf-mod48-ratio-test-boundary","seed":202611172088,"semantic_template":"ratio-test-boundary|{mode:\"zero_terms\",scale:14}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=zero_terms and positive integer scale=14. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(zero_terms, 14, a_n=14_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0089","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aabd88f93d44c6a0d4f47af3da1f72d8a948aec1f504a13da95d3f42d0988318","family_id":"qf-mod48-ratio-test-boundary","seed":202611172089,"semantic_template":"ratio-test-boundary|{mode:\"finite_samples\",scale:15}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=finite_samples and positive integer scale=15. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(finite_samples, 15, ratios_equal_1/2_for_n<=15_then_equal_2, finite_prefix_cannot_determine_ratio_limit, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0090","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63934f044ba8cf845ff7b8598b9a84f72d382d139e34c690a37fc858457de8f4","family_id":"qf-mod48-ratio-test-boundary","seed":202611172090,"semantic_template":"ratio-test-boundary|{mode:\"missing_limit\",scale:16}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=missing_limit and positive integer scale=16. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(missing_limit, 16, absolute_ratios_alternate_between_1/17_and_17, limit_does_not_exist, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0091","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65b93df547ce14097c5aaa54a5e6d8d719b5b0e9ebf8be4c845ce35ec70187cb","family_id":"qf-mod48-ratio-test-boundary","seed":202611172091,"semantic_template":"ratio-test-boundary|{mode:\"signed_without_absolute\",scale:17}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=signed_without_absolute and positive integer scale=17. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(signed_without_absolute, 17, a_n=(-1)^n*17*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0092","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bab76f1785d900be64cf676223d0d74541c94c870039705e5908b3743861c828","family_id":"qf-mod48-ratio-test-boundary","seed":202611172092,"semantic_template":"ratio-test-boundary|{mode:\"zero_terms\",scale:18}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=zero_terms and positive integer scale=18. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(zero_terms, 18, a_n=18_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0093","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3cbb88db5aa0eb52b30d21d002efc6347a2a2962621a4e5b1a5c78fb0cdc192","family_id":"qf-mod48-ratio-test-boundary","seed":202611172093,"semantic_template":"ratio-test-boundary|{mode:\"finite_samples\",scale:19}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=finite_samples and positive integer scale=19. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(finite_samples, 19, ratios_equal_1/2_for_n<=19_then_equal_2, finite_prefix_cannot_determine_ratio_limit, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0094","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7265580d6c2c78e3c1a4b3ca4bf2c074481cbdc1189b01c687e5e694264ae5b","family_id":"qf-mod48-ratio-test-boundary","seed":202611172094,"semantic_template":"ratio-test-boundary|{mode:\"missing_limit\",scale:20}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=missing_limit and positive integer scale=20. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(missing_limit, 20, absolute_ratios_alternate_between_1/21_and_21, limit_does_not_exist, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0095","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0005ac7ea2c0905783e3e79d75ac6245b2c093bc230a3dd61ebd87b5d98b71ed","family_id":"qf-mod48-ratio-test-boundary","seed":202611172095,"semantic_template":"ratio-test-boundary|{mode:\"signed_without_absolute\",scale:21}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=signed_without_absolute and positive integer scale=21. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(signed_without_absolute, 21, a_n=(-1)^n*21*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0096","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"accb5dd1e7afb1cbaa50f8b8c2d3929989199656ed5d5b57b9342008908619be","family_id":"qf-mod48-ratio-test-boundary","seed":202611172096,"semantic_template":"ratio-test-boundary|{mode:\"zero_terms\",scale:22}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=zero_terms and positive integer scale=22. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(zero_terms, 22, a_n=22_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0097","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93bf9177db56f31016d480746b4bd2c19258f843f08ba745ad05cd054c945dfe","family_id":"qf-mod48-ratio-test-boundary","seed":202611172097,"semantic_template":"ratio-test-boundary|{mode:\"finite_samples\",scale:23}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=finite_samples and positive integer scale=23. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(finite_samples, 23, ratios_equal_1/2_for_n<=23_then_equal_2, finite_prefix_cannot_determine_ratio_limit, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0098","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"977802e69241d68f567b11a6afe8519c4aab1451ec38cb68b4cce80725464f66","family_id":"qf-mod48-ratio-test-boundary","seed":202611172098,"semantic_template":"ratio-test-boundary|{mode:\"missing_limit\",scale:24}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=missing_limit and positive integer scale=24. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(missing_limit, 24, absolute_ratios_alternate_between_1/25_and_25, limit_does_not_exist, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0099","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86d246a259d8c14c6f39ce55826c94a286cb81ee9e423c34c87cadec198c86e2","family_id":"qf-mod48-ratio-test-boundary","seed":202611172099,"semantic_template":"ratio-test-boundary|{mode:\"signed_without_absolute\",scale:25}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=signed_without_absolute and positive integer scale=25. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(signed_without_absolute, 25, a_n=(-1)^n*25*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r02-ratio-test-series-0100","batch_id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0c883f85c9c701c0d0d846072c25e03f2dee44f727d8f25034e29617d4f896d","family_id":"qf-mod48-ratio-test-boundary","seed":202611172100,"semantic_template":"ratio-test-boundary|{mode:\"zero_terms\",scale:26}"},"fields":{"title":"Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises","stem":"Use mode=zero_terms and positive integer scale=26. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).","answer":{"kind":"expression","value":"(zero_terms, 26, a_n=26_for_odd_n_and_0_for_even_n, successive_ratio_has_zero_denominators_infinitely_often, No)","explanation":"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)."},"skills":["Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}