{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0001","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5c69f1480bffbcb45ae9af6912c6496fce92045eba9606293c259109318d161","family_id":"qf-mod43-radius-geometric-exact","seed":202611125001,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-12,\"coefficient_ratio\":[1,2]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(1/2)^n for every n≥0, positive ratio q=1/2, and center c=-12. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(1/2)^n_for_every_n>=0, -12, 2, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(1/2)^n_for_every_n>=0, -12, 2, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0002","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aab2fe604f11c27a1b7e8ba91070fa8f3dffbfbe37540c0b65193074df55f7c5","family_id":"qf-mod43-radius-geometric-exact","seed":202611125002,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-11,\"coefficient_ratio\":[2,3]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(2/3)^n for every n≥0, positive ratio q=2/3, and center c=-11. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(2/3)^n_for_every_n>=0, -11, 3/2, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(2/3)^n_for_every_n>=0, -11, 3/2, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0003","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f77c4c14201b2d2003258697258bea93c80d3b487cda6eaef1633d5f08c26286","family_id":"qf-mod43-radius-geometric-exact","seed":202611125003,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-10,\"coefficient_ratio\":[3,4]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(3/4)^n for every n≥0, positive ratio q=3/4, and center c=-10. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(3/4)^n_for_every_n>=0, -10, 4/3, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(3/4)^n_for_every_n>=0, -10, 4/3, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0004","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"785e9e1d49b1c5d652d2f7550d15898ba626b1069314fd20cd3992df1201d99f","family_id":"qf-mod43-radius-geometric-exact","seed":202611125004,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-9,\"coefficient_ratio\":[4,5]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(4/5)^n for every n≥0, positive ratio q=4/5, and center c=-9. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(4/5)^n_for_every_n>=0, -9, 5/4, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(4/5)^n_for_every_n>=0, -9, 5/4, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0005","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"646dabb7684e3344b937f1f8924ae23476cb19f4611b63d8ef1f7eed42cfad56","family_id":"qf-mod43-radius-geometric-exact","seed":202611125005,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-8,\"coefficient_ratio\":[5,6]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(5/6)^n for every n≥0, positive ratio q=5/6, and center c=-8. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(5/6)^n_for_every_n>=0, -8, 6/5, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(5/6)^n_for_every_n>=0, -8, 6/5, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0006","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f212dce61ba1fe8f5d0012f9acd3d5e6b7c873dba1c13bf754a2934d9a9fe76","family_id":"qf-mod43-radius-geometric-exact","seed":202611125006,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-7,\"coefficient_ratio\":[6,7]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(6/7)^n for every n≥0, positive ratio q=6/7, and center c=-7. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(6/7)^n_for_every_n>=0, -7, 7/6, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(6/7)^n_for_every_n>=0, -7, 7/6, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0007","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17b51181930ed25bd4115b8e27f3b6e967ff9f2c0ed07ab65b4d743590e512d3","family_id":"qf-mod43-radius-geometric-exact","seed":202611125007,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-6,\"coefficient_ratio\":[7,8]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(7/8)^n for every n≥0, positive ratio q=7/8, and center c=-6. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(7/8)^n_for_every_n>=0, -6, 8/7, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(7/8)^n_for_every_n>=0, -6, 8/7, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0008","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afbf1b96a19b329166544140f79f8fde40fb974202bea5890c470fa62464c649","family_id":"qf-mod43-radius-geometric-exact","seed":202611125008,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-5,\"coefficient_ratio\":[8,9]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(8/9)^n for every n≥0, positive ratio q=8/9, and center c=-5. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(8/9)^n_for_every_n>=0, -5, 9/8, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(8/9)^n_for_every_n>=0, -5, 9/8, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0009","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3922d7ed1a85a3b2b81382c45481c06340109a20a3574f940928d30929985ca","family_id":"qf-mod43-radius-geometric-exact","seed":202611125009,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-4,\"coefficient_ratio\":[9,10]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(9/10)^n for every n≥0, positive ratio q=9/10, and center c=-4. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(9/10)^n_for_every_n>=0, -4, 10/9, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(9/10)^n_for_every_n>=0, -4, 10/9, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0010","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74c5f3fd0c90cd122c4bbe416497e684051efad42fdb9c239a1381b9793728b5","family_id":"qf-mod43-radius-geometric-exact","seed":202611125010,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-3,\"coefficient_ratio\":[10,11]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(10/11)^n for every n≥0, positive ratio q=10/11, and center c=-3. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(10/11)^n_for_every_n>=0, -3, 11/10, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(10/11)^n_for_every_n>=0, -3, 11/10, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0011","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"490a54d7154e61258e1b36fd05f037b96009c856b2389014ab60d5bc604ec93b","family_id":"qf-mod43-radius-geometric-exact","seed":202611125011,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-2,\"coefficient_ratio\":[11,12]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(11/12)^n for every n≥0, positive ratio q=11/12, and center c=-2. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(11/12)^n_for_every_n>=0, -2, 12/11, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(11/12)^n_for_every_n>=0, -2, 12/11, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0012","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fd91dc7728edba9dd00699a123aa55e35ed16627be7bc6563afb4d788d26e72","family_id":"qf-mod43-radius-geometric-exact","seed":202611125012,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-1,\"coefficient_ratio\":[12,13]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(12/13)^n for every n≥0, positive ratio q=12/13, and center c=-1. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(12/13)^n_for_every_n>=0, -1, 13/12, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(12/13)^n_for_every_n>=0, -1, 13/12, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0013","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e7161ae062ed344395d9ee2c5efad8f809be106918aa9bcdc4d510dca6be072","family_id":"qf-mod43-radius-geometric-exact","seed":202611125013,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":0,\"coefficient_ratio\":[13,14]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(13/14)^n for every n≥0, positive ratio q=13/14, and center c=0. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(13/14)^n_for_every_n>=0, 0, 14/13, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(13/14)^n_for_every_n>=0, 0, 14/13, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0014","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c628e30c36f6f353812184b06d21efd5b3bb3e7ed94365c52f2573ae70f37e35","family_id":"qf-mod43-radius-geometric-exact","seed":202611125014,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":1,\"coefficient_ratio\":[14,15]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(14/15)^n for every n≥0, positive ratio q=14/15, and center c=1. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(14/15)^n_for_every_n>=0, 1, 15/14, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(14/15)^n_for_every_n>=0, 1, 15/14, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0015","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b9da184bab5e056899a6bc363df5b28e7c687234da581ae7541312ff3c505d1","family_id":"qf-mod43-radius-geometric-exact","seed":202611125015,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":2,\"coefficient_ratio\":[15,16]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(15/16)^n for every n≥0, positive ratio q=15/16, and center c=2. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(15/16)^n_for_every_n>=0, 2, 16/15, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(15/16)^n_for_every_n>=0, 2, 16/15, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0016","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09a4866f9f61a8060520651f7356c8a46c83eaf351988dbd8f12416e28698efb","family_id":"qf-mod43-radius-geometric-exact","seed":202611125016,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":3,\"coefficient_ratio\":[16,17]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(16/17)^n for every n≥0, positive ratio q=16/17, and center c=3. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(16/17)^n_for_every_n>=0, 3, 17/16, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(16/17)^n_for_every_n>=0, 3, 17/16, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0017","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ec623e7538f6650a4d39607083fcda455892cd440c2e13f2c1d4d9475e83c44","family_id":"qf-mod43-radius-geometric-exact","seed":202611125017,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":4,\"coefficient_ratio\":[17,18]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(17/18)^n for every n≥0, positive ratio q=17/18, and center c=4. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(17/18)^n_for_every_n>=0, 4, 18/17, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(17/18)^n_for_every_n>=0, 4, 18/17, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0018","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40f0bb608e73853eecab9c2b6c2494b52bb34bcb680bda019413b35e4e912cf2","family_id":"qf-mod43-radius-geometric-exact","seed":202611125018,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":5,\"coefficient_ratio\":[18,19]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(18/19)^n for every n≥0, positive ratio q=18/19, and center c=5. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(18/19)^n_for_every_n>=0, 5, 19/18, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(18/19)^n_for_every_n>=0, 5, 19/18, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0019","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8312050583a13b2ef745775ac48bb4fa171a5137132cde01db1e2bde692b5698","family_id":"qf-mod43-radius-geometric-exact","seed":202611125019,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":6,\"coefficient_ratio\":[19,20]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(19/20)^n for every n≥0, positive ratio q=19/20, and center c=6. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(19/20)^n_for_every_n>=0, 6, 20/19, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(19/20)^n_for_every_n>=0, 6, 20/19, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0020","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75df7735c0e4528e6a47f224fb6b2292b138d5b3e1cac2c0d8a878a69cb7a0cb","family_id":"qf-mod43-radius-geometric-exact","seed":202611125020,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":7,\"coefficient_ratio\":[20,21]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(20/21)^n for every n≥0, positive ratio q=20/21, and center c=7. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(20/21)^n_for_every_n>=0, 7, 21/20, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(20/21)^n_for_every_n>=0, 7, 21/20, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0021","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36870646e3c8645ecc41dab21dddf3af2a2bf9603b7922606f0d2189a83725c8","family_id":"qf-mod43-radius-geometric-exact","seed":202611125021,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":8,\"coefficient_ratio\":[21,22]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(21/22)^n for every n≥0, positive ratio q=21/22, and center c=8. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(21/22)^n_for_every_n>=0, 8, 22/21, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(21/22)^n_for_every_n>=0, 8, 22/21, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0022","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73dbe3536d9aa9cebd26b1edf8f2c74dbb4f4bf7630260653b670806cefad4fe","family_id":"qf-mod43-radius-geometric-exact","seed":202611125022,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":9,\"coefficient_ratio\":[22,23]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(22/23)^n for every n≥0, positive ratio q=22/23, and center c=9. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(22/23)^n_for_every_n>=0, 9, 23/22, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(22/23)^n_for_every_n>=0, 9, 23/22, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0023","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37c8f110d5f7b993849db4d2cf118e664b499aaebf01ebf0a573f39b5247091a","family_id":"qf-mod43-radius-geometric-exact","seed":202611125023,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":10,\"coefficient_ratio\":[23,24]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(23/24)^n for every n≥0, positive ratio q=23/24, and center c=10. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(23/24)^n_for_every_n>=0, 10, 24/23, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(23/24)^n_for_every_n>=0, 10, 24/23, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0024","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b7b00c4e7c58ec22d9840682a9e03c93afeb9966c2d71dde59ed413be058d02","family_id":"qf-mod43-radius-geometric-exact","seed":202611125024,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":11,\"coefficient_ratio\":[24,25]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(24/25)^n for every n≥0, positive ratio q=24/25, and center c=11. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(24/25)^n_for_every_n>=0, 11, 25/24, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(24/25)^n_for_every_n>=0, 11, 25/24, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0025","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b89fd8b95b6f2908564c00aa56bc20cd064e4f17dd9c651c227084e5484f9860","family_id":"qf-mod43-radius-geometric-exact","seed":202611125025,"semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":12,\"coefficient_ratio\":[25,26]}"},"fields":{"title":"A Whole Geometric Coefficient Law Gives an Exact Radius","stem":"Use a_n=(25/26)^n for every n≥0, positive ratio q=25/26, and center c=12. Give (coefficient_law, center, radius, interior_status, exterior_status, boundary).","answer":{"kind":"expression","value":"(a_n=(25/26)^n_for_every_n>=0, 12, 26/25, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation":"For sum a_n*(x−c)^n the radius is1/q. Interior absolute convergence and exterior divergence are distinct from endpoints, which need a separate audit. A finite coefficient sample is not the complete law. The exact result, retaining the requested order, is (a_n=(25/26)^n_for_every_n>=0, 12, 26/25, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)."},"skills":["A Whole Geometric Coefficient Law Gives an Exact Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0026","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"458c234775312c14f4c99fe61a7202d2c7da5636d1b790f142a52f8d1d021001","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125026,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-12,\"coefficient_ratio\":[1,2],\"point\":-12}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(1/2)^n for every n≥0, positive q=1/2, center=-12, point=-12. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(1/2)^n_for_every_n>=0, -12, -12, 0, 2, absolute_convergence, Yes, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(1/2)^n_for_every_n>=0, -12, -12, 0, 2, absolute_convergence, Yes, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0027","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13a95246e9fdad8c1d8b39fd06e74c6907e6b87d7a0c4a7c99f5b5e2889f7d4f","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125027,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-11,\"coefficient_ratio\":[2,3],\"point\":-9}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(2/3)^n for every n≥0, positive q=2/3, center=-11, point=-9. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(2/3)^n_for_every_n>=0, -11, -9, 2, 3/2, divergence, No, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(2/3)^n_for_every_n>=0, -11, -9, 2, 3/2, divergence, No, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0028","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01b371317d6a7d613acdec79c065f45c6a11837d97d8582104869583982de077","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125028,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-10,\"coefficient_ratio\":[3,4],\"point\":-10}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(3/4)^n for every n≥0, positive q=3/4, center=-10, point=-10. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(3/4)^n_for_every_n>=0, -10, -10, 0, 4/3, absolute_convergence, Yes, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(3/4)^n_for_every_n>=0, -10, -10, 0, 4/3, absolute_convergence, Yes, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0029","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d734d23462d5a5d5de8cccb0ec3e744f0c42bec695937ec31460e01e6dc87e4","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125029,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-9,\"coefficient_ratio\":[4,5],\"point\":-7}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(4/5)^n for every n≥0, positive q=4/5, center=-9, point=-7. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(4/5)^n_for_every_n>=0, -9, -7, 2, 5/4, divergence, No, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(4/5)^n_for_every_n>=0, -9, -7, 2, 5/4, divergence, No, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0030","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f741d313a11e1eac7be57ed3e24d00184d70613535488740f1e5a705cceb895","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125030,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-8,\"coefficient_ratio\":[5,6],\"point\":-8}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(5/6)^n for every n≥0, positive q=5/6, center=-8, point=-8. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(5/6)^n_for_every_n>=0, -8, -8, 0, 6/5, absolute_convergence, Yes, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(5/6)^n_for_every_n>=0, -8, -8, 0, 6/5, absolute_convergence, Yes, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0031","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a943e0a5797ae082c5895f38b1541df90fbfb78c0b9a50840453fcb07053eafd","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125031,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-7,\"coefficient_ratio\":[6,7],\"point\":-5}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(6/7)^n for every n≥0, positive q=6/7, center=-7, point=-5. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(6/7)^n_for_every_n>=0, -7, -5, 2, 7/6, divergence, No, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(6/7)^n_for_every_n>=0, -7, -5, 2, 7/6, divergence, No, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0032","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6689118305a1ae4e45a8c245de2b45a95746884348d89cc8bb625efe75a9d999","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125032,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-6,\"coefficient_ratio\":[7,8],\"point\":-6}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(7/8)^n for every n≥0, positive q=7/8, center=-6, point=-6. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(7/8)^n_for_every_n>=0, -6, -6, 0, 8/7, absolute_convergence, Yes, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(7/8)^n_for_every_n>=0, -6, -6, 0, 8/7, absolute_convergence, Yes, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0033","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32cf14c3274e0949ee4168625e8c1e5e0425968d8e398c83733045b002082e1e","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125033,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-5,\"coefficient_ratio\":[8,9],\"point\":-3}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(8/9)^n for every n≥0, positive q=8/9, center=-5, point=-3. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(8/9)^n_for_every_n>=0, -5, -3, 2, 9/8, divergence, No, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(8/9)^n_for_every_n>=0, -5, -3, 2, 9/8, divergence, No, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0034","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ceb236fdfbe243b9fa383b0d774602c3f1c0e43cf3618cc460ce76cc0859058","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125034,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-4,\"coefficient_ratio\":[9,10],\"point\":-4}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(9/10)^n for every n≥0, positive q=9/10, center=-4, point=-4. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(9/10)^n_for_every_n>=0, -4, -4, 0, 10/9, absolute_convergence, Yes, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(9/10)^n_for_every_n>=0, -4, -4, 0, 10/9, absolute_convergence, Yes, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0035","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ebe3bb2288b99084eb16a642cafdfe1ae840ec4a5d6bc11151c88fef64698a7","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125035,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-3,\"coefficient_ratio\":[10,11],\"point\":-1}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(10/11)^n for every n≥0, positive q=10/11, center=-3, point=-1. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(10/11)^n_for_every_n>=0, -3, -1, 2, 11/10, divergence, No, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(10/11)^n_for_every_n>=0, -3, -1, 2, 11/10, divergence, No, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0036","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7db121a650eed3797e3baf7168a4bacb0e3821e5e000117d1b99ec2ecbce68e","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125036,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-2,\"coefficient_ratio\":[11,12],\"point\":-2}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(11/12)^n for every n≥0, positive q=11/12, center=-2, point=-2. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(11/12)^n_for_every_n>=0, -2, -2, 0, 12/11, absolute_convergence, Yes, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(11/12)^n_for_every_n>=0, -2, -2, 0, 12/11, absolute_convergence, Yes, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0037","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50360f59baa9e01675e792e68405f82065badab9da91fa405f518f2d7c2c211b","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125037,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-1,\"coefficient_ratio\":[12,13],\"point\":1}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(12/13)^n for every n≥0, positive q=12/13, center=-1, point=1. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(12/13)^n_for_every_n>=0, -1, 1, 2, 13/12, divergence, No, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(12/13)^n_for_every_n>=0, -1, 1, 2, 13/12, divergence, No, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0038","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0974dde04fac8db1dd351354ab31be50faf7ac140c1d372012e386481918d9a8","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125038,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":0,\"coefficient_ratio\":[13,14],\"point\":0}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(13/14)^n for every n≥0, positive q=13/14, center=0, point=0. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(13/14)^n_for_every_n>=0, 0, 0, 0, 14/13, absolute_convergence, Yes, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(13/14)^n_for_every_n>=0, 0, 0, 0, 14/13, absolute_convergence, Yes, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0039","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20c36efc262288d705eda83b031cbf8b6f80105545f5a372146b36e5c44026c3","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125039,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":1,\"coefficient_ratio\":[14,15],\"point\":3}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(14/15)^n for every n≥0, positive q=14/15, center=1, point=3. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(14/15)^n_for_every_n>=0, 1, 3, 2, 15/14, divergence, No, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(14/15)^n_for_every_n>=0, 1, 3, 2, 15/14, divergence, No, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0040","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12453f511f01a622d03db36ba931ab5dccd7a9e8ab9240f48161ebd21050c422","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125040,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":2,\"coefficient_ratio\":[15,16],\"point\":2}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(15/16)^n for every n≥0, positive q=15/16, center=2, point=2. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(15/16)^n_for_every_n>=0, 2, 2, 0, 16/15, absolute_convergence, Yes, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(15/16)^n_for_every_n>=0, 2, 2, 0, 16/15, absolute_convergence, Yes, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0041","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1cbbbbb39f74ca10db87a20852244b06d8b129d44d92e275d02a2531dd5bdeb","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125041,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":3,\"coefficient_ratio\":[16,17],\"point\":5}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(16/17)^n for every n≥0, positive q=16/17, center=3, point=5. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(16/17)^n_for_every_n>=0, 3, 5, 2, 17/16, divergence, No, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(16/17)^n_for_every_n>=0, 3, 5, 2, 17/16, divergence, No, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0042","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15c3b641514f63fdcffe7d1abd8b54ab909d649dc3d4541e8f5107db508be829","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125042,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":4,\"coefficient_ratio\":[17,18],\"point\":4}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(17/18)^n for every n≥0, positive q=17/18, center=4, point=4. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(17/18)^n_for_every_n>=0, 4, 4, 0, 18/17, absolute_convergence, Yes, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(17/18)^n_for_every_n>=0, 4, 4, 0, 18/17, absolute_convergence, Yes, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0043","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18aea9207320665576cbf74ee6c0036fcff77ab863350e2ae6102b3ac54847f0","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125043,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":5,\"coefficient_ratio\":[18,19],\"point\":7}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(18/19)^n for every n≥0, positive q=18/19, center=5, point=7. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(18/19)^n_for_every_n>=0, 5, 7, 2, 19/18, divergence, No, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(18/19)^n_for_every_n>=0, 5, 7, 2, 19/18, divergence, No, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0044","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"beb42fac96dbdd6e804e4621b36528ffcf3dbad50bbdb655f5d2f808e9a4bcdd","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125044,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":6,\"coefficient_ratio\":[19,20],\"point\":6}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(19/20)^n for every n≥0, positive q=19/20, center=6, point=6. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(19/20)^n_for_every_n>=0, 6, 6, 0, 20/19, absolute_convergence, Yes, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(19/20)^n_for_every_n>=0, 6, 6, 0, 20/19, absolute_convergence, Yes, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0045","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51dec6cf740f4666bc4a4c8c15ec76716a27a23852f32013ab3ec31d963f1c33","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125045,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":7,\"coefficient_ratio\":[20,21],\"point\":9}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(20/21)^n for every n≥0, positive q=20/21, center=7, point=9. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(20/21)^n_for_every_n>=0, 7, 9, 2, 21/20, divergence, No, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(20/21)^n_for_every_n>=0, 7, 9, 2, 21/20, divergence, No, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0046","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a12842b805ba046eca88419194cf2a05bdf8d487d7d23b45b6106bbfe0a14188","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125046,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":8,\"coefficient_ratio\":[21,22],\"point\":8}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(21/22)^n for every n≥0, positive q=21/22, center=8, point=8. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(21/22)^n_for_every_n>=0, 8, 8, 0, 22/21, absolute_convergence, Yes, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(21/22)^n_for_every_n>=0, 8, 8, 0, 22/21, absolute_convergence, Yes, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0047","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2421e7f821ff5d40786f7feb03cfafa03fda7bf1b282bb2ca39b4840c8751b5","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125047,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":9,\"coefficient_ratio\":[22,23],\"point\":11}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(22/23)^n for every n≥0, positive q=22/23, center=9, point=11. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(22/23)^n_for_every_n>=0, 9, 11, 2, 23/22, divergence, No, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(22/23)^n_for_every_n>=0, 9, 11, 2, 23/22, divergence, No, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0048","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5275a86215726e1195aaa90563c1c5e652378c1d556121b69ec3d7bbc97f2e70","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125048,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":10,\"coefficient_ratio\":[23,24],\"point\":10}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(23/24)^n for every n≥0, positive q=23/24, center=10, point=10. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(23/24)^n_for_every_n>=0, 10, 10, 0, 24/23, absolute_convergence, Yes, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(23/24)^n_for_every_n>=0, 10, 10, 0, 24/23, absolute_convergence, Yes, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0049","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5bb04986f2055a644a853556c50d9bd372d0966e5c6cbc55a8a90fc0675d29c","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125049,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":11,\"coefficient_ratio\":[24,25],\"point\":13}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(24/25)^n for every n≥0, positive q=24/25, center=11, point=13. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(24/25)^n_for_every_n>=0, 11, 13, 2, 25/24, divergence, No, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(24/25)^n_for_every_n>=0, 11, 13, 2, 25/24, divergence, No, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0050","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65912517e6a8ff955509315c4c833cb6fa92b22583c0e09946b5fcb4c21310ec","family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125050,"semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":12,\"coefficient_ratio\":[25,26],\"point\":12}"},"fields":{"title":"Classify the Exact Distance from the Declared Center","stem":"Use a_n=(25/26)^n for every n≥0, positive q=25/26, center=12, point=12. Give (coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(a_n=(25/26)^n_for_every_n>=0, 12, 12, 0, 26/25, absolute_convergence, Yes, classification_uses_exact_distance_from_center)","explanation":"Compare |point−center| with exact radius1/q. This family excludes equality: smaller distance gives absolute convergence, larger gives divergence. Do not silently include either endpoint. The exact result, retaining the requested order, is (a_n=(25/26)^n_for_every_n>=0, 12, 12, 0, 26/25, absolute_convergence, Yes, classification_uses_exact_distance_from_center)."},"skills":["Classify the Exact Distance from the Declared Center"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0051","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72851bb9ce9902cbddff5722569e361b7655b52b0d8137973a582219fb816181","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125051,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-12,\"decay_power\":0,\"radius\":1}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=1^(−n)/(n+1)^0 for every n≥0 and center=-12, with radius R=1>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=1^(-n)/(n+1)^0_for_every_n>=0, -12, 1, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=1^(-n)/(n+1)^0_for_every_n>=0, -12, 1, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0052","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecb7910d4b75683f83208a22135ba0f2216921617b4a5f2a78bf3fd2ae46f884","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125052,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-11,\"decay_power\":1,\"radius\":2}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=2^(−n)/(n+1)^1 for every n≥0 and center=-11, with radius R=2>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=2^(-n)/(n+1)^1_for_every_n>=0, -11, 2, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=2^(-n)/(n+1)^1_for_every_n>=0, -11, 2, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0053","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9bf3cf592c99892890f66185a0ba28460b9aa05036c9e1b4ba2f938edabd011a","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125053,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-10,\"decay_power\":2,\"radius\":3}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=3^(−n)/(n+1)^2 for every n≥0 and center=-10, with radius R=3>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=3^(-n)/(n+1)^2_for_every_n>=0, -10, 3, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=3^(-n)/(n+1)^2_for_every_n>=0, -10, 3, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0054","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bc147aae0cbebea29c76a050f13f0a59d9b033fca793b582be5888ead3a3eb0","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125054,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-9,\"decay_power\":0,\"radius\":4}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=4^(−n)/(n+1)^0 for every n≥0 and center=-9, with radius R=4>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=4^(-n)/(n+1)^0_for_every_n>=0, -9, 4, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=4^(-n)/(n+1)^0_for_every_n>=0, -9, 4, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0055","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ba6a27d829683946666bd241f213c85c4f264790312fba6919fa10d1003f5a5","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125055,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-8,\"decay_power\":1,\"radius\":5}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=5^(−n)/(n+1)^1 for every n≥0 and center=-8, with radius R=5>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=5^(-n)/(n+1)^1_for_every_n>=0, -8, 5, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=5^(-n)/(n+1)^1_for_every_n>=0, -8, 5, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0056","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5102d3b935d8bb7de4dd401d17d7c8aae62fadf8e780912554df80f751be00ef","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125056,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-7,\"decay_power\":2,\"radius\":1}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=1^(−n)/(n+1)^2 for every n≥0 and center=-7, with radius R=1>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=1^(-n)/(n+1)^2_for_every_n>=0, -7, 1, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=1^(-n)/(n+1)^2_for_every_n>=0, -7, 1, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0057","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94eab21f8849ace1e388ab755610594ab6584184c8aefd1d8a049730d4f78bcc","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125057,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-6,\"decay_power\":0,\"radius\":2}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=2^(−n)/(n+1)^0 for every n≥0 and center=-6, with radius R=2>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=2^(-n)/(n+1)^0_for_every_n>=0, -6, 2, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=2^(-n)/(n+1)^0_for_every_n>=0, -6, 2, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0058","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"578aa891be19dc1b9bbee85dfd65802a554b2373888c08f277d66d785f390d91","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125058,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-5,\"decay_power\":1,\"radius\":3}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=3^(−n)/(n+1)^1 for every n≥0 and center=-5, with radius R=3>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=3^(-n)/(n+1)^1_for_every_n>=0, -5, 3, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=3^(-n)/(n+1)^1_for_every_n>=0, -5, 3, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0059","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af57f1fd2d484f61d6d545eb4c937b2379233e72788044bb56f64ce4d834d3b7","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125059,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-4,\"decay_power\":2,\"radius\":4}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=4^(−n)/(n+1)^2 for every n≥0 and center=-4, with radius R=4>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=4^(-n)/(n+1)^2_for_every_n>=0, -4, 4, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=4^(-n)/(n+1)^2_for_every_n>=0, -4, 4, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0060","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5138ddcea8ce38fdda0fa8d3e5296b4037c20e0b08df20fb6a4d78a023cc4a10","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125060,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-3,\"decay_power\":0,\"radius\":5}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=5^(−n)/(n+1)^0 for every n≥0 and center=-3, with radius R=5>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=5^(-n)/(n+1)^0_for_every_n>=0, -3, 5, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=5^(-n)/(n+1)^0_for_every_n>=0, -3, 5, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0061","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe9d9bab96b02dcd6547ffdfe303fd343983f2f372997e94b9017bb7ab5e0abd","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125061,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-2,\"decay_power\":1,\"radius\":1}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=1^(−n)/(n+1)^1 for every n≥0 and center=-2, with radius R=1>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=1^(-n)/(n+1)^1_for_every_n>=0, -2, 1, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=1^(-n)/(n+1)^1_for_every_n>=0, -2, 1, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0062","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36e47a1fde37d5e13c17d4ccb72517bc447944d39cd2e71b3f016d6916f50c21","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125062,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-1,\"decay_power\":2,\"radius\":2}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=2^(−n)/(n+1)^2 for every n≥0 and center=-1, with radius R=2>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=2^(-n)/(n+1)^2_for_every_n>=0, -1, 2, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=2^(-n)/(n+1)^2_for_every_n>=0, -1, 2, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0063","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0001f5acdb52cabb32983874ede0131174a695761c2c357f3a82c9248688c66f","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125063,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":0,\"decay_power\":0,\"radius\":3}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=3^(−n)/(n+1)^0 for every n≥0 and center=0, with radius R=3>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=3^(-n)/(n+1)^0_for_every_n>=0, 0, 3, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=3^(-n)/(n+1)^0_for_every_n>=0, 0, 3, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0064","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a3616ffb9c66ad268b69493b53688aaead0b65b8f070f13b6a165fc04405ecd","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125064,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":1,\"decay_power\":1,\"radius\":4}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=4^(−n)/(n+1)^1 for every n≥0 and center=1, with radius R=4>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=4^(-n)/(n+1)^1_for_every_n>=0, 1, 4, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=4^(-n)/(n+1)^1_for_every_n>=0, 1, 4, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0065","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22348d962651426e6a46118ea65554938c355ac99c5f9de976c72c94e3499300","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125065,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":2,\"decay_power\":2,\"radius\":5}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=5^(−n)/(n+1)^2 for every n≥0 and center=2, with radius R=5>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=5^(-n)/(n+1)^2_for_every_n>=0, 2, 5, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=5^(-n)/(n+1)^2_for_every_n>=0, 2, 5, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0066","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c4c688e536a7dab31e49d47dfdd11d3ca307fce21d2a11f0042586cc95cf6d7","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125066,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":3,\"decay_power\":0,\"radius\":1}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=1^(−n)/(n+1)^0 for every n≥0 and center=3, with radius R=1>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=1^(-n)/(n+1)^0_for_every_n>=0, 3, 1, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=1^(-n)/(n+1)^0_for_every_n>=0, 3, 1, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0067","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3a5740399c1bd354df3512668e752baf5eda0ac493725a0a998c29be046f1b7","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125067,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":4,\"decay_power\":1,\"radius\":2}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=2^(−n)/(n+1)^1 for every n≥0 and center=4, with radius R=2>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=2^(-n)/(n+1)^1_for_every_n>=0, 4, 2, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=2^(-n)/(n+1)^1_for_every_n>=0, 4, 2, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0068","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92c1d9294bd9190c11d1e2c0efc95f9d440c65956f3252eeb5f46d0ee5049381","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125068,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":5,\"decay_power\":2,\"radius\":3}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=3^(−n)/(n+1)^2 for every n≥0 and center=5, with radius R=3>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=3^(-n)/(n+1)^2_for_every_n>=0, 5, 3, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=3^(-n)/(n+1)^2_for_every_n>=0, 5, 3, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0069","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57ca71bc7146e05fe17ccb83c3a2a26ae2526fd4a1bae2dc761013302dfa77a6","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125069,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":6,\"decay_power\":0,\"radius\":4}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=4^(−n)/(n+1)^0 for every n≥0 and center=6, with radius R=4>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=4^(-n)/(n+1)^0_for_every_n>=0, 6, 4, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=4^(-n)/(n+1)^0_for_every_n>=0, 6, 4, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0070","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfcb5d19e1627a47be5f21af169aa04a9ce0145db9db20a50d6d15a9bc1cbe36","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125070,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":7,\"decay_power\":1,\"radius\":5}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=5^(−n)/(n+1)^1 for every n≥0 and center=7, with radius R=5>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=5^(-n)/(n+1)^1_for_every_n>=0, 7, 5, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=5^(-n)/(n+1)^1_for_every_n>=0, 7, 5, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0071","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ca0270a2ed5a708e69f19198f3179d88fcc1a409458895197e9c09fcf126963","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125071,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":8,\"decay_power\":2,\"radius\":1}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=1^(−n)/(n+1)^2 for every n≥0 and center=8, with radius R=1>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=1^(-n)/(n+1)^2_for_every_n>=0, 8, 1, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=1^(-n)/(n+1)^2_for_every_n>=0, 8, 1, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0072","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76866750095d256a944c6b63c9b3aaef8a022195c9ca270beac346a7cfef3a13","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125072,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":9,\"decay_power\":0,\"radius\":2}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=2^(−n)/(n+1)^0 for every n≥0 and center=9, with radius R=2>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=2^(-n)/(n+1)^0_for_every_n>=0, 9, 2, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=2^(-n)/(n+1)^0_for_every_n>=0, 9, 2, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0073","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49a665bb16336166d80ea5b60e0277bbfd166faff9acb80bc55f6900970afa77","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125073,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":10,\"decay_power\":1,\"radius\":3}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=3^(−n)/(n+1)^1 for every n≥0 and center=10, with radius R=3>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=3^(-n)/(n+1)^1_for_every_n>=0, 10, 3, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=3^(-n)/(n+1)^1_for_every_n>=0, 10, 3, 1, divergence, conditional_convergence, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0074","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f1ae5f453f9c3b859918ba123cbc78d219e1134acd516b141c80b71740b28e7","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125074,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":11,\"decay_power\":2,\"radius\":4}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=4^(−n)/(n+1)^2 for every n≥0 and center=11, with radius R=4>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=4^(-n)/(n+1)^2_for_every_n>=0, 11, 4, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=4^(-n)/(n+1)^2_for_every_n>=0, 11, 4, 2, absolute_convergence, absolute_convergence, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0075","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84720cfce7e658c577e32faffb50d70a3d3611f8e6de2250bb421cacc7933eb0","family_id":"qf-mod43-radius-endpoint-independent","seed":202611125075,"semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":12,\"decay_power\":0,\"radius\":5}"},"fields":{"title":"The Positive and Negative Endpoints Can Have Different Status under the Same Radius","stem":"Use a_n=5^(−n)/(n+1)^0 for every n≥0 and center=12, with radius R=5>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).","answer":{"kind":"expression","value":"(a_n=5^(-n)/(n+1)^0_for_every_n>=0, 12, 5, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)","explanation":"At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=5^(-n)/(n+1)^0_for_every_n>=0, 12, 5, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)."},"skills":["The Positive and Negative Endpoints Can Have Different Status under the Same Radius"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0076","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27d8b006ae63b87a9cb416f3438bff0df516b4340943e09e384fb139e65779e4","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125076,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"zero_radius\",\"prefix_length\":0,\"scale\":1,\"second_ratio\":1}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=zero_radius, positive scale=1, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(zero_radius, 1, a_n=1*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (zero_radius, 1, a_n=1*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0077","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"219d1574d0808f75cc8fd2c64fa4900260b406a5487dd8c85712a7c4a25e9ac9","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125077,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"infinite_radius\",\"prefix_length\":0,\"scale\":2,\"second_ratio\":1}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=infinite_radius, positive scale=2, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(infinite_radius, 2, a_n=2/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (infinite_radius, 2, a_n=2/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0078","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d64652ee8975fac21fd5b83eb64bb17e3b15b87648d0002f87bce71b82c3902","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125078,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":[1,2],\"mode\":\"finite_prefix\",\"prefix_length\":1,\"scale\":3,\"second_ratio\":[1,3]}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=finite_prefix, positive scale=3, source prefix_length=1, first_ratio=1/2, second_ratio=1/3. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate).","answer":{"kind":"expression","value":"(finite_prefix, 3, 1, [0, 0], a_n=0_for_n<=1;a_n=3*(1/2)^(n-2)_for_n>1, b_n=0_for_n<=1;b_n=3*(1/3)^(n-2)_for_n>1, 2, 3, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (finite_prefix, 3, 1, [0, 0], a_n=0_for_n<=1;a_n=3*(1/2)^(n-2)_for_n>1, b_n=0_for_n<=1;b_n=3*(1/3)^(n-2)_for_n>1, 2, 3, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0079","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"829351681d22bada4ab78c024bf5b72b2ca97e4d581f6c56e1fff294f44b5b81","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125079,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"zero_radius\",\"prefix_length\":0,\"scale\":4,\"second_ratio\":1}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=zero_radius, positive scale=4, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(zero_radius, 4, a_n=4*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (zero_radius, 4, a_n=4*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0080","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e4914bf6df3badd1a44928f4f762c1bb3446c6a2bf953a910fb58d67863b7c1","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125080,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"infinite_radius\",\"prefix_length\":0,\"scale\":5,\"second_ratio\":1}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=infinite_radius, positive scale=5, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(infinite_radius, 5, a_n=5/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (infinite_radius, 5, a_n=5/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0081","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a4b41e59c78875f4a01a5252877c78822a4b3bd1f1f9ff6e3f0de38ec7c616b","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125081,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":[1,2],\"mode\":\"finite_prefix\",\"prefix_length\":2,\"scale\":6,\"second_ratio\":[1,4]}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=finite_prefix, positive scale=6, source prefix_length=2, first_ratio=1/2, second_ratio=1/4. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate).","answer":{"kind":"expression","value":"(finite_prefix, 6, 2, [0, 0, 0], a_n=0_for_n<=2;a_n=6*(1/2)^(n-3)_for_n>2, b_n=0_for_n<=2;b_n=6*(1/4)^(n-3)_for_n>2, 2, 4, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (finite_prefix, 6, 2, [0, 0, 0], a_n=0_for_n<=2;a_n=6*(1/2)^(n-3)_for_n>2, b_n=0_for_n<=2;b_n=6*(1/4)^(n-3)_for_n>2, 2, 4, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0082","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffc1a6a03b537e165e6516223fc2439dea41e6372f6f44af5938921c3d3679a7","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125082,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"zero_radius\",\"prefix_length\":0,\"scale\":7,\"second_ratio\":1}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=zero_radius, positive scale=7, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(zero_radius, 7, a_n=7*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (zero_radius, 7, a_n=7*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0083","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2774102fd41fe6e94c88acaf7fe3a387466171db42c9a81e32d7c71901eeb65","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125083,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"infinite_radius\",\"prefix_length\":0,\"scale\":8,\"second_ratio\":1}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=infinite_radius, positive scale=8, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(infinite_radius, 8, a_n=8/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (infinite_radius, 8, a_n=8/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0084","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d812ccb91a28ae961df9c77b2dbf6f8bc715429851172fdaa3867d16b5ba76b8","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125084,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":[1,2],\"mode\":\"finite_prefix\",\"prefix_length\":3,\"scale\":9,\"second_ratio\":[1,5]}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=finite_prefix, positive scale=9, source prefix_length=3, first_ratio=1/2, second_ratio=1/5. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate).","answer":{"kind":"expression","value":"(finite_prefix, 9, 3, [0, 0, 0, 0], a_n=0_for_n<=3;a_n=9*(1/2)^(n-4)_for_n>3, b_n=0_for_n<=3;b_n=9*(1/5)^(n-4)_for_n>3, 2, 5, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (finite_prefix, 9, 3, [0, 0, 0, 0], a_n=0_for_n<=3;a_n=9*(1/2)^(n-4)_for_n>3, b_n=0_for_n<=3;b_n=9*(1/5)^(n-4)_for_n>3, 2, 5, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0085","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccdffe4060e10105aee5737cf7b2c984206d1c40a6d1684a047f10829771a560","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125085,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"zero_radius\",\"prefix_length\":0,\"scale\":10,\"second_ratio\":1}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=zero_radius, positive scale=10, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(zero_radius, 10, a_n=10*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (zero_radius, 10, a_n=10*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0086","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6006f926015620cffd85bffa707dbf79c7825c1795509517654ce3dbdfdedce","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125086,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"infinite_radius\",\"prefix_length\":0,\"scale\":11,\"second_ratio\":1}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=infinite_radius, positive scale=11, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(infinite_radius, 11, a_n=11/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (infinite_radius, 11, a_n=11/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0087","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d1ee90d6b45f53982274a0baa2054f7f1676cb2589ad1d21ebc04f4c204b761","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125087,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":[1,2],\"mode\":\"finite_prefix\",\"prefix_length\":4,\"scale\":12,\"second_ratio\":[1,6]}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=finite_prefix, positive scale=12, source prefix_length=4, first_ratio=1/2, second_ratio=1/6. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate).","answer":{"kind":"expression","value":"(finite_prefix, 12, 4, [0, 0, 0, 0, 0], a_n=0_for_n<=4;a_n=12*(1/2)^(n-5)_for_n>4, b_n=0_for_n<=4;b_n=12*(1/6)^(n-5)_for_n>4, 2, 6, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (finite_prefix, 12, 4, [0, 0, 0, 0, 0], a_n=0_for_n<=4;a_n=12*(1/2)^(n-5)_for_n>4, b_n=0_for_n<=4;b_n=12*(1/6)^(n-5)_for_n>4, 2, 6, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0088","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"209948afbcc0bb59a5c2c5738087c96578ab55e184398a22e7577f15f77cd619","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125088,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"zero_radius\",\"prefix_length\":0,\"scale\":13,\"second_ratio\":1}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=zero_radius, positive scale=13, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(zero_radius, 13, a_n=13*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (zero_radius, 13, a_n=13*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0089","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b981f5c25390197e9dfa52802259551cfa161e8f77cd104de58ddbf3b4ae54c5","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125089,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"infinite_radius\",\"prefix_length\":0,\"scale\":14,\"second_ratio\":1}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=infinite_radius, positive scale=14, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(infinite_radius, 14, a_n=14/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (infinite_radius, 14, a_n=14/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0090","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d95b8830a8ac59f7aae4061504b43582b6eceb6e04ee19922f2387c025b3450","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125090,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":[1,2],\"mode\":\"finite_prefix\",\"prefix_length\":5,\"scale\":15,\"second_ratio\":[1,7]}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=finite_prefix, positive scale=15, source prefix_length=5, first_ratio=1/2, second_ratio=1/7. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate).","answer":{"kind":"expression","value":"(finite_prefix, 15, 5, [0, 0, 0, 0, 0, 0], a_n=0_for_n<=5;a_n=15*(1/2)^(n-6)_for_n>5, b_n=0_for_n<=5;b_n=15*(1/7)^(n-6)_for_n>5, 2, 7, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (finite_prefix, 15, 5, [0, 0, 0, 0, 0, 0], a_n=0_for_n<=5;a_n=15*(1/2)^(n-6)_for_n>5, b_n=0_for_n<=5;b_n=15*(1/7)^(n-6)_for_n>5, 2, 7, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0091","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd1968407764081382fcec6b5035b1332c05d96fb967fff373f421ead7d8ccb7","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125091,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"zero_radius\",\"prefix_length\":0,\"scale\":16,\"second_ratio\":1}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=zero_radius, positive scale=16, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(zero_radius, 16, a_n=16*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (zero_radius, 16, a_n=16*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0092","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e694530b6a76746ce47e5a616db7fde87d53cab72b10840a817c3161ae6691f3","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125092,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"infinite_radius\",\"prefix_length\":0,\"scale\":17,\"second_ratio\":1}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=infinite_radius, positive scale=17, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(infinite_radius, 17, a_n=17/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (infinite_radius, 17, a_n=17/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0093","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2f07740cef23f152a8d2d74075f04c4c05e7deb72551d148034e7cb4699bcca","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125093,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":[1,2],\"mode\":\"finite_prefix\",\"prefix_length\":6,\"scale\":18,\"second_ratio\":[1,8]}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=finite_prefix, positive scale=18, source prefix_length=6, first_ratio=1/2, second_ratio=1/8. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate).","answer":{"kind":"expression","value":"(finite_prefix, 18, 6, [0, 0, 0, 0, 0, 0, 0], a_n=0_for_n<=6;a_n=18*(1/2)^(n-7)_for_n>6, b_n=0_for_n<=6;b_n=18*(1/8)^(n-7)_for_n>6, 2, 8, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (finite_prefix, 18, 6, [0, 0, 0, 0, 0, 0, 0], a_n=0_for_n<=6;a_n=18*(1/2)^(n-7)_for_n>6, b_n=0_for_n<=6;b_n=18*(1/8)^(n-7)_for_n>6, 2, 8, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0094","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1844d2d04f48c60c88388d6ccb58fdaaae2b11448181cccbda0582edc61bd47","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125094,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"zero_radius\",\"prefix_length\":0,\"scale\":19,\"second_ratio\":1}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=zero_radius, positive scale=19, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(zero_radius, 19, a_n=19*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (zero_radius, 19, a_n=19*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0095","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d862fd51be388b74abdd8b4f55c8e88d445d49984374662a45be5e84dfa8ca93","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125095,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"infinite_radius\",\"prefix_length\":0,\"scale\":20,\"second_ratio\":1}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=infinite_radius, positive scale=20, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(infinite_radius, 20, a_n=20/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (infinite_radius, 20, a_n=20/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0096","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"faf85e9a5c3a0ef226d38239c9009d92be3e29231f7b7dcd3f9e50119257980e","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125096,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":[1,2],\"mode\":\"finite_prefix\",\"prefix_length\":7,\"scale\":21,\"second_ratio\":[1,9]}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=finite_prefix, positive scale=21, source prefix_length=7, first_ratio=1/2, second_ratio=1/9. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate).","answer":{"kind":"expression","value":"(finite_prefix, 21, 7, [0, 0, 0, 0, 0, 0, 0, 0], a_n=0_for_n<=7;a_n=21*(1/2)^(n-8)_for_n>7, b_n=0_for_n<=7;b_n=21*(1/9)^(n-8)_for_n>7, 2, 9, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (finite_prefix, 21, 7, [0, 0, 0, 0, 0, 0, 0, 0], a_n=0_for_n<=7;a_n=21*(1/2)^(n-8)_for_n>7, b_n=0_for_n<=7;b_n=21*(1/9)^(n-8)_for_n>7, 2, 9, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0097","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b254fe07453286cb8b5d0a8be8bd328b64515afc2fcd8e88757e18e70e1c49a9","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125097,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"zero_radius\",\"prefix_length\":0,\"scale\":22,\"second_ratio\":1}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=zero_radius, positive scale=22, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(zero_radius, 22, a_n=22*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (zero_radius, 22, a_n=22*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0098","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e706a7133b26c9d4ced07ea516ab1019516de6910f53453a8564802152728727","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125098,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"infinite_radius\",\"prefix_length\":0,\"scale\":23,\"second_ratio\":1}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=infinite_radius, positive scale=23, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(infinite_radius, 23, a_n=23/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (infinite_radius, 23, a_n=23/n!_for_every_n>=0, infinity, Yes, factorial_denominator_gives_infinite_radius)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0099","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"259f68264c68c7cce110a41dd4e06c5b03ba4134612ed7d437bbeeb84f1bbb29","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125099,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":[1,2],\"mode\":\"finite_prefix\",\"prefix_length\":8,\"scale\":24,\"second_ratio\":[1,10]}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=finite_prefix, positive scale=24, source prefix_length=8, first_ratio=1/2, second_ratio=1/10. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate).","answer":{"kind":"expression","value":"(finite_prefix, 24, 8, [0, 0, 0, 0, 0, 0, 0, 0, 0], a_n=0_for_n<=8;a_n=24*(1/2)^(n-9)_for_n>8, b_n=0_for_n<=8;b_n=24*(1/10)^(n-9)_for_n>8, 2, 10, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (finite_prefix, 24, 8, [0, 0, 0, 0, 0, 0, 0, 0, 0], a_n=0_for_n<=8;a_n=24*(1/2)^(n-9)_for_n>8, b_n=0_for_n<=8;b_n=24*(1/10)^(n-9)_for_n>8, 2, 10, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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-20261014-advance-20261112-r05-power-series-radius-of-convergence-0100","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b23ac5eb583a42ee581e50eeeb717f8e768fe8da1dcc6cabf8d060e531f3d71e","family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125100,"semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"zero_radius\",\"prefix_length\":0,\"scale\":25,\"second_ratio\":1}"},"fields":{"title":"Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii","stem":"Use mode=zero_radius, positive scale=25, source prefix_length=0, first_ratio=1, second_ratio=1. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, coefficient_law, radius, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(zero_radius, 25, a_n=25*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)","explanation":"zero_radius declares a_n=scale*n! for every n≥0 and R=0; infinite_radius declares a_n=scale/n! and R=infinity. finite_prefix uses N=source prefix_length as the last shared zero index, so there are N+1 zeros; subsequent coefficients are scale*q_1^(n−N−1) or scale*q_2^(n−N−1), with positive unequal q values and radii1/q_1,1/q_2. Use the explicit mode-selected6 or11 fields without padding or answer-shape inference. The exact result, retaining the requested order, is (zero_radius, 25, a_n=25*n!_for_every_n>=0, 0, Yes, factorial_growth_gives_radius_zero)."},"skills":["Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii"],"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."}}