{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261104-r03-absolute-convergence-of-series-0001","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"074ce292068dc8734386696d0ecfdf78d31b1a8ac67098ed7e96e2979ba83d60","family_id":"qf-mod35-absseries-geometric","seed":202611043001,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[1,2],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=1/2, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(1/2, -1/2, 1, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (1/2, -1/2, 1, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0002","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f27cc7f508d5db78aef523de77f2e0444f17ab921fb9000cdcb2cd1f8920b679","family_id":"qf-mod35-absseries-geometric","seed":202611043002,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[2,3],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=2/3, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(2/3, -1/2, 4/3, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (2/3, -1/2, 4/3, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0003","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"647daeff7039e06e326dd7661fbceb002f038fb2e089a4cb85d3e79767eda834","family_id":"qf-mod35-absseries-geometric","seed":202611043003,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[3,4],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=3/4, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(3/4, -1/2, 3/2, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (3/4, -1/2, 3/2, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0004","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"505279f240da0daeef85dade1ba02b47dfceb592858592e1e1895e6b65061960","family_id":"qf-mod35-absseries-geometric","seed":202611043004,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[4,5],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=4/5, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(4/5, -1/2, 8/5, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (4/5, -1/2, 8/5, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0005","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7af45d0c5fc85aa6b482d3833bcb2dd98c74a72284614aa29e0e7adcdc922328","family_id":"qf-mod35-absseries-geometric","seed":202611043005,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[5,6],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=5/6, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(5/6, -1/2, 5/3, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (5/6, -1/2, 5/3, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0006","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f0162ebcfe41fe206df2dd192ce598b4f6f697cc7e5ef2db2414b46abad6be9","family_id":"qf-mod35-absseries-geometric","seed":202611043006,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[6,7],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=6/7, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(6/7, -1/2, 12/7, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (6/7, -1/2, 12/7, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0007","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e9d7857af5ea4b1a55d1a29ebc16091c4b2e11666be61a5f70ef759ed8ba29a","family_id":"qf-mod35-absseries-geometric","seed":202611043007,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[7,8],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=7/8, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(7/8, -1/2, 7/4, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (7/8, -1/2, 7/4, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0008","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11d3f4911dac0662a08eb1f0a4d1c6dcfc13b8a07e55edd125cafef11e65f53d","family_id":"qf-mod35-absseries-geometric","seed":202611043008,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[8,9],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=8/9, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(8/9, -1/2, 16/9, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (8/9, -1/2, 16/9, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0009","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73df20755f41d05093665e387652a1fe1ce19c6dec1978f7ea07883173241279","family_id":"qf-mod35-absseries-geometric","seed":202611043009,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[9,10],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=9/10, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(9/10, -1/2, 9/5, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (9/10, -1/2, 9/5, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0010","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23df7c411a90b1f8013b5e6a0e15765c6167084e138cac348c21cd16921b2e81","family_id":"qf-mod35-absseries-geometric","seed":202611043010,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[10,11],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=10/11, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(10/11, -1/2, 20/11, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (10/11, -1/2, 20/11, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0011","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f527194d9e92adc9e11ba20a078338b33a331c5c1012f0ad688477cff43583c","family_id":"qf-mod35-absseries-geometric","seed":202611043011,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[11,12],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=11/12, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(11/12, -1/2, 11/6, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (11/12, -1/2, 11/6, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0012","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0c537207fcc1b802e4b36f6d3c90725c6598aafa482295b7d8fd937b06cb2c4","family_id":"qf-mod35-absseries-geometric","seed":202611043012,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[12,13],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=12/13, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(12/13, -1/2, 24/13, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (12/13, -1/2, 24/13, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0013","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5440852fd4f243afcbd18fe81a409a8312865b11a1661276aca8b3b4b57c96db","family_id":"qf-mod35-absseries-geometric","seed":202611043013,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[13,14],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=13/14, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(13/14, -1/2, 13/7, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (13/14, -1/2, 13/7, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0014","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1bb0e864bddc4bad8d0d002cce29b09bbb70b4c539c35e89d9db8ac8f784864","family_id":"qf-mod35-absseries-geometric","seed":202611043014,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[14,15],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=14/15, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(14/15, -1/2, 28/15, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (14/15, -1/2, 28/15, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0015","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24f5d7e1dbaff7c44abdc9e2d8f8024ef61c96df0a18d634d516404c05fb6375","family_id":"qf-mod35-absseries-geometric","seed":202611043015,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[15,16],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=15/16, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(15/16, -1/2, 15/8, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (15/16, -1/2, 15/8, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0016","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97ea59dd39d82a31054e309df6ade99835380a0c9cf92a47cf0179e414a8ed64","family_id":"qf-mod35-absseries-geometric","seed":202611043016,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[16,17],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=16/17, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(16/17, -1/2, 32/17, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (16/17, -1/2, 32/17, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0017","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d46b2bc6333de75f989108fe8579235eee653757381093011bf174a20cefcaf","family_id":"qf-mod35-absseries-geometric","seed":202611043017,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[17,18],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=17/18, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(17/18, -1/2, 17/9, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (17/18, -1/2, 17/9, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0018","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09be9955b00aed110d8aa92fd7814d7c6ce171bd7eb39681fbfcdddc08242fb2","family_id":"qf-mod35-absseries-geometric","seed":202611043018,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[18,19],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=18/19, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(18/19, -1/2, 36/19, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (18/19, -1/2, 36/19, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0019","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e300a62760761911ac66fe27486f0f64a38a5ff600c3c6c23433cef11e1e51b9","family_id":"qf-mod35-absseries-geometric","seed":202611043019,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[19,20],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=19/20, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(19/20, -1/2, 19/10, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (19/20, -1/2, 19/10, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0020","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"428c55585cd5f84b10ec4df06aa513cb96fa8596e62b84dc6cee89243ed0ff42","family_id":"qf-mod35-absseries-geometric","seed":202611043020,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[20,21],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=20/21, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(20/21, -1/2, 40/21, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (20/21, -1/2, 40/21, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0021","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"822f15d16a3872c7dfe589d98672fbe2995a61d3e1ac33906c5681116aafc8df","family_id":"qf-mod35-absseries-geometric","seed":202611043021,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[21,22],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=21/22, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(21/22, -1/2, 21/11, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (21/22, -1/2, 21/11, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0022","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f1dc9b49fba97d4feaf94494555b86f0d0d4e8b7e0d13b9a588d1840da4a517","family_id":"qf-mod35-absseries-geometric","seed":202611043022,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[22,23],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=22/23, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(22/23, -1/2, 44/23, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (22/23, -1/2, 44/23, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0023","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cb34315ffcf9f8850bf770bb65cb9d6a1370923bf0476661ded6a324e1dc902","family_id":"qf-mod35-absseries-geometric","seed":202611043023,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[23,24],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=23/24, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(23/24, -1/2, 23/12, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (23/24, -1/2, 23/12, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0024","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2f93be7bceee875798cc17c2343138f1310624dcf58d42fa18f23e10d916b09","family_id":"qf-mod35-absseries-geometric","seed":202611043024,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[24,25],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=24/25, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(24/25, -1/2, 48/25, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (24/25, -1/2, 48/25, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0025","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aac6cb88c52f171fd1e2f4e235f7c18f7d6d4a0ccfeb56d0c9ceedab5a62a92f","family_id":"qf-mod35-absseries-geometric","seed":202611043025,"semantic_template":"qf-mod35-absseries-geometric/{\"a\":[25,26],\"r\":[-1,2]}"},"fields":{"title":"Sum Absolute Terms, Not the Absolute Value of the Signed Sum","stem":"Use terms a*r^k for k=0,1,... with exact a=25/26, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).","answer":{"kind":"expression","value":"(25/26, -1/2, 25/13, Yes, absolute_geometric_certificate)","explanation":"The sum of absolute terms is |a|/(1−|r|). Retain the signed a and r input fields separately from that nonnegative sum. The explicit geometric tail supplies absolute convergence and hence convergence of the original series, not a finite-prefix inference. The exact result, retaining the requested order, is (25/26, -1/2, 25/13, Yes, absolute_geometric_certificate)."},"skills":["Sum Absolute Terms, Not the Absolute Value of the Signed Sum"],"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-20261104-r03-absolute-convergence-of-series-0026","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecedbfe32b719e7bf4f6c8879ea0ebcc2855e7bf6913cf7e8188996cd8f3d241","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043026,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":1}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=1, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(1, 5, [1, -1/2, 1/3, -1/4, 1/5], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (1, 5, [1, -1/2, 1/3, -1/4, 1/5], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0027","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec6f3dbb201e73cce603c304c16ffdc3291caf736e12314bf5b43857a5cd324e","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043027,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":2}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=2, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(2, 5, [1/2, -1/3, 1/4, -1/5, 1/6], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (2, 5, [1/2, -1/3, 1/4, -1/5, 1/6], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0028","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3666fd6a8508555a5bd8b9233d7b281e8ef7b5912542f94a9b68d2aefd26d648","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043028,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":3}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=3, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(3, 5, [1/3, -1/4, 1/5, -1/6, 1/7], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (3, 5, [1/3, -1/4, 1/5, -1/6, 1/7], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0029","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63d351f56d9f5c02b9522b271b12802b9bc01aa949eaa8a5082efb2d6826a736","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043029,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":4}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=4, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(4, 5, [1/4, -1/5, 1/6, -1/7, 1/8], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (4, 5, [1/4, -1/5, 1/6, -1/7, 1/8], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0030","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"769174f583cb40adab34fe55845df3c9e0eb103ddbb5113007b0079cfe02c04a","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043030,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":5}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=5, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(5, 5, [1/5, -1/6, 1/7, -1/8, 1/9], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (5, 5, [1/5, -1/6, 1/7, -1/8, 1/9], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0031","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f188817fc0578228bd4f98f5cd6705fb3b7d1c33eaec6962e952d9791878694","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043031,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":6}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=6, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(6, 5, [1/6, -1/7, 1/8, -1/9, 1/10], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (6, 5, [1/6, -1/7, 1/8, -1/9, 1/10], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0032","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73834f872890c08a83835f33e9e4cb29e57eec6a41f97a8efb435f176eb08685","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043032,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":7}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=7, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(7, 5, [1/7, -1/8, 1/9, -1/10, 1/11], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (7, 5, [1/7, -1/8, 1/9, -1/10, 1/11], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0033","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"671273651f65d6d4e1ffda302070670bd84775dc7eb9ea1e355b771600efb899","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043033,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":8}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=8, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(8, 5, [1/8, -1/9, 1/10, -1/11, 1/12], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (8, 5, [1/8, -1/9, 1/10, -1/11, 1/12], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0034","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67d322b89b52769ac1a100b199b9fdc3e508a1d8c811c38fd24218e7892af09b","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043034,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":9}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=9, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(9, 5, [1/9, -1/10, 1/11, -1/12, 1/13], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (9, 5, [1/9, -1/10, 1/11, -1/12, 1/13], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0035","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26e02556489d1fdad5719e180dd84b000babcc2488253d4b7fef0cedd568ef96","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043035,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":10}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=10, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(10, 5, [1/10, -1/11, 1/12, -1/13, 1/14], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (10, 5, [1/10, -1/11, 1/12, -1/13, 1/14], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0036","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9413ce704ea014e67134c08eb0e4c47ee3d3f3161a0043599e8a6312e29095fc","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043036,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":11}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=11, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(11, 5, [1/11, -1/12, 1/13, -1/14, 1/15], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (11, 5, [1/11, -1/12, 1/13, -1/14, 1/15], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0037","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39a94650546562b7e4e639ad05f52f1980ea82eb520915d72601a26baf5b8c55","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043037,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":12}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=12, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(12, 5, [1/12, -1/13, 1/14, -1/15, 1/16], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (12, 5, [1/12, -1/13, 1/14, -1/15, 1/16], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0038","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"803725ae7ec880a4c899b1fbee4cab52c7101fe334201379f025e98038fba138","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043038,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":13}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=13, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(13, 5, [1/13, -1/14, 1/15, -1/16, 1/17], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (13, 5, [1/13, -1/14, 1/15, -1/16, 1/17], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0039","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6968464f1f69109d8d0400d9022c8938bd6789f38b90cb0a3ad8ba1226ccd0ea","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043039,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":14}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=14, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(14, 5, [1/14, -1/15, 1/16, -1/17, 1/18], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (14, 5, [1/14, -1/15, 1/16, -1/17, 1/18], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0040","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07f47063bf02e2a3303c5b7ec77791ab8f922525e16881b9033bf8a297845e11","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043040,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":15}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=15, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(15, 5, [1/15, -1/16, 1/17, -1/18, 1/19], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (15, 5, [1/15, -1/16, 1/17, -1/18, 1/19], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0041","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e64ff89e44979b61e7cd83f820e6c81456858e6115ff76a33799b8e980ed52d","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043041,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":16}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=16, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(16, 5, [1/16, -1/17, 1/18, -1/19, 1/20], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (16, 5, [1/16, -1/17, 1/18, -1/19, 1/20], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0042","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcaa078e2ac75df903405f53cefbdae87dd218735a51068bfe3b2a72b5d14a3d","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043042,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":17}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=17, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(17, 5, [1/17, -1/18, 1/19, -1/20, 1/21], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (17, 5, [1/17, -1/18, 1/19, -1/20, 1/21], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0043","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb5878bde2f288c0de53806486576e846d796fa937c61a16934d313a1ae59d84","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043043,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":18}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=18, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(18, 5, [1/18, -1/19, 1/20, -1/21, 1/22], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (18, 5, [1/18, -1/19, 1/20, -1/21, 1/22], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0044","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82995dec21a7095702311942f6daf1ab58bfa3b3033ad091e8b561af46d2edd0","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043044,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":19}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=19, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(19, 5, [1/19, -1/20, 1/21, -1/22, 1/23], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (19, 5, [1/19, -1/20, 1/21, -1/22, 1/23], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0045","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9765f985c2534cf328b7e79eebe19e51ed133ae1ef3e6c36422a88c586c42b7a","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043045,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":20}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=20, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(20, 5, [1/20, -1/21, 1/22, -1/23, 1/24], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (20, 5, [1/20, -1/21, 1/22, -1/23, 1/24], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0046","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"381406911539b4d02b78b2fc425a4e374de7b536c6b2e794f1d10e423c624ae5","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043046,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":21}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=21, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(21, 5, [1/21, -1/22, 1/23, -1/24, 1/25], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (21, 5, [1/21, -1/22, 1/23, -1/24, 1/25], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0047","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8046b9fe07e4664e02667c5816d213235f5a89482962d095f24e24a1028892df","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043047,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":22}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=22, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(22, 5, [1/22, -1/23, 1/24, -1/25, 1/26], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (22, 5, [1/22, -1/23, 1/24, -1/25, 1/26], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0048","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8193b0a60142e3e74774537291f1941306b8e4b29faa344dfe29e24611f2a872","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043048,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":23}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=23, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(23, 5, [1/23, -1/24, 1/25, -1/26, 1/27], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (23, 5, [1/23, -1/24, 1/25, -1/26, 1/27], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0049","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9033d8412e54cb0b12c7bd2ea3946e430d22fca22ad49239913b2ebda68bf4f7","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043049,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":24}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=24, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(24, 5, [1/24, -1/25, 1/26, -1/27, 1/28], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (24, 5, [1/24, -1/25, 1/26, -1/27, 1/28], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0050","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88df4e636eaf172d6d5ff19c7a03ef8367f3900a3ee1c2dd091c495ff8a00b62","family_id":"qf-mod35-absseries-conditional-alternating","seed":202611043050,"semantic_template":"qf-mod35-absseries-conditional-alternating/{\"n\":5,\"shift\":25}"},"fields":{"title":"Conditional Convergence Needs Both Convergence and Absolute Divergence","stem":"Use terms (−1)^k/(k+shift), k=0,1,..., positive shift=25, and display n=5 terms. Give (shift, n, terms, series_converges, absolutely_converges, certificate).","answer":{"kind":"expression","value":"(25, 5, [1/25, -1/26, 1/27, -1/28, 1/29], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)","explanation":"The alternating remainder argument certifies convergence of this declared whole series, while divergence of the shifted harmonic absolute series certifies failure of absolute convergence. The displayed finite list alone is neither proof. The exact result, retaining the requested order, is (25, 5, [1/25, -1/26, 1/27, -1/28, 1/29], Yes, No, alternating_remainder_and_harmonic_absolute_divergence)."},"skills":["Conditional Convergence Needs Both Convergence and Absolute Divergence"],"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-20261104-r03-absolute-convergence-of-series-0051","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fda49737532fb177701834d197294d71b79b9d8c465c7c3210a59a4e7f17a38","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043051,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,2],[-1,3]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/2, -1/3]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/2, -1/3], [1/2, 1/3], 5/6, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/2, -1/3], [1/2, 1/3], 5/6, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0052","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01b5b7af4720720b15d67ac4668f640da503338796d691aa95606cd3e2ecfcdb","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043052,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,3],[-1,4]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/3, -1/4]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/3, -1/4], [1/3, 1/4], 7/12, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/3, -1/4], [1/3, 1/4], 7/12, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0053","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f170e6189361b7bc2b065afae152cf8f58f19f11a96e8798bf716a2556a3f1c4","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043053,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,4],[-1,5]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/4, -1/5]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/4, -1/5], [1/4, 1/5], 9/20, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/4, -1/5], [1/4, 1/5], 9/20, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0054","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b37de0e12e4e77050551fde256ae2bb7b1fc6f14202163786f16a4f9e1afc89a","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043054,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,5],[-1,6]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/5, -1/6]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/5, -1/6], [1/5, 1/6], 11/30, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/5, -1/6], [1/5, 1/6], 11/30, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0055","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6e8621860513715dff589504887824eda71d1b4e1eef21c7cdd325050588edd","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043055,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,6],[-1,7]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/6, -1/7]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/6, -1/7], [1/6, 1/7], 13/42, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/6, -1/7], [1/6, 1/7], 13/42, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0056","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"907de859d4f39f0c8c288ef100a1b1b3b2442a04b7831119731e94d1128595df","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043056,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,7],[-1,8]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/7, -1/8]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/7, -1/8], [1/7, 1/8], 15/56, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/7, -1/8], [1/7, 1/8], 15/56, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0057","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6a6e65028e5307eac6565e38264509c11541883faf89cabb634c43c40f33450","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043057,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,8],[-1,9]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/8, -1/9]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/8, -1/9], [1/8, 1/9], 17/72, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/8, -1/9], [1/8, 1/9], 17/72, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0058","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9a1f93d90b5649fb6481cac11482fd308a1510ee877f61fe1b9c34feaeef9b5","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043058,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,9],[-1,10]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/9, -1/10]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/9, -1/10], [1/9, 1/10], 19/90, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/9, -1/10], [1/9, 1/10], 19/90, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0059","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7585323e2431611cf7ceef6af91dfa77808d815d4c69d87d3b755d4155bf677","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043059,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,10],[-1,11]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/10, -1/11]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/10, -1/11], [1/10, 1/11], 21/110, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/10, -1/11], [1/10, 1/11], 21/110, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0060","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a2580bfed858cf4d9c5523d4452adcc12c7d8cd309d76dbf3277bf13647fcea","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043060,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,11],[-1,12]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/11, -1/12]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/11, -1/12], [1/11, 1/12], 23/132, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/11, -1/12], [1/11, 1/12], 23/132, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0061","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd949f391a9b22c8fefe07050fe5d042c2a046ee08dff995abacd7cc4a9134cc","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043061,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,12],[-1,13]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/12, -1/13]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/12, -1/13], [1/12, 1/13], 25/156, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/12, -1/13], [1/12, 1/13], 25/156, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0062","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7e5a19a357ed00f85c6c3f137e9f9b91bd074d115c28c8ab6b1cf77ebd7488f","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043062,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,13],[-1,14]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/13, -1/14]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/13, -1/14], [1/13, 1/14], 27/182, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/13, -1/14], [1/13, 1/14], 27/182, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0063","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0b95e68958ed0af04b8399c9c739ea0c610d886849a16f4ac63cd9ddea69108","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043063,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,14],[-1,15]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/14, -1/15]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/14, -1/15], [1/14, 1/15], 29/210, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/14, -1/15], [1/14, 1/15], 29/210, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0064","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cab135b665e7fd768b20a5028bf7339a9efd76b2aac54cfeac49bf341da0fe7","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043064,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,15],[-1,16]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/15, -1/16]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/15, -1/16], [1/15, 1/16], 31/240, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/15, -1/16], [1/15, 1/16], 31/240, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0065","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"785d72ed2bbabc3e7c989ba193bf0f6d9b7f90567805e579feb4d536b79caf86","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043065,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,16],[-1,17]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/16, -1/17]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/16, -1/17], [1/16, 1/17], 33/272, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/16, -1/17], [1/16, 1/17], 33/272, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0066","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e0e7b6876cb6cc04fc50835e4c8a55ee459289bc32e7b8c4497cf9248f65bb3","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043066,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,17],[-1,18]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/17, -1/18]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/17, -1/18], [1/17, 1/18], 35/306, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/17, -1/18], [1/17, 1/18], 35/306, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0067","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed266ed43612277b78297a3cb63766d411c3e79cdefd3fd5386ad0bce4dc54f0","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043067,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,18],[-1,19]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/18, -1/19]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/18, -1/19], [1/18, 1/19], 37/342, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/18, -1/19], [1/18, 1/19], 37/342, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0068","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec90f0839383c0e3f273838723871291648b12076e3c766648a791c59627fca8","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043068,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,19],[-1,20]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/19, -1/20]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/19, -1/20], [1/19, 1/20], 39/380, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/19, -1/20], [1/19, 1/20], 39/380, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0069","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a5c5c8bcbbfdc960ac529c0ae426dd027bd8a882307d5fee9b202726d118d9f","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043069,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,20],[-1,21]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/20, -1/21]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/20, -1/21], [1/20, 1/21], 41/420, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/20, -1/21], [1/20, 1/21], 41/420, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0070","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f73341bb81390e5d58ac8b224f81d561728446eb475394d59ce241e2556cdd63","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043070,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,21],[-1,22]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/21, -1/22]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/21, -1/22], [1/21, 1/22], 43/462, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/21, -1/22], [1/21, 1/22], 43/462, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0071","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"114ab46c30f8d7c55371591161bb72595ce32f430eb958e8c02501b3ed6a2755","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043071,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,22],[-1,23]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/22, -1/23]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/22, -1/23], [1/22, 1/23], 45/506, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/22, -1/23], [1/22, 1/23], 45/506, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0072","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e56b86ed40e2babfa50f40a9d5c752696c80b073cf94a36177d1b88f11b502b9","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043072,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,23],[-1,24]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/23, -1/24]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/23, -1/24], [1/23, 1/24], 47/552, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/23, -1/24], [1/23, 1/24], 47/552, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0073","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e9252c043414908949c151b8fe6cf68186071ab01bf1468358fc67190b33832","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043073,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,24],[-1,25]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/24, -1/25]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/24, -1/25], [1/24, 1/25], 49/600, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/24, -1/25], [1/24, 1/25], 49/600, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0074","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b44852898140dee7b5a38b33049be3c2938d09aaeed207c7d21e45be25139b4b","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043074,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,25],[-1,26]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/25, -1/26]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/25, -1/26], [1/25, 1/26], 51/650, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/25, -1/26], [1/25, 1/26], 51/650, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0075","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"beb37b7b73fa31957e33c383b041ddfd607b26d012fce60a30b546b6ddfddb3f","family_id":"qf-mod35-absseries-absolute-partial","seed":202611043075,"semantic_template":"qf-mod35-absseries-absolute-partial/{\"terms\":[[1,26],[-1,27]]}"},"fields":{"title":"A Finite Sum of Absolute Values Is Not an Infinite Certificate","stem":"Supplied ordered exact rational terms=[1/26, -1/27]. Give (terms, absolute_terms, absolute_partial_sum, boundary).","answer":{"kind":"expression","value":"([1/26, -1/27], [1/26, 1/27], 53/702, finite_absolute_partial_sum_only)","explanation":"Take each absolute value, then add exactly. Keep the original signed term list and the absolute term list in the same order. This finite sum is not the absolute value of the signed sum or a proof about an unspecified infinite continuation. The exact result, retaining the requested order, is ([1/26, -1/27], [1/26, 1/27], 53/702, finite_absolute_partial_sum_only)."},"skills":["A Finite Sum of Absolute Values Is Not an Infinite Certificate"],"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-20261104-r03-absolute-convergence-of-series-0076","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2b9e3ef97e94fb6f23f2341d1af13cfdf3fed2741c183ddec23c0dac7a25adc","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043076,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":1}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=1 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(1, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (1, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0077","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce415cf668fa27e192424faa0875afbc28c72f398be7b0cf7f1384b11713b893","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043077,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":2}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=2 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(2, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (2, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0078","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"534c5b58c709885986aaf660fc480354bf2ee86b7ebc4fa168221f0004fcd8a5","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043078,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":3}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=3 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(3, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (3, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0079","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c6d57e0a99cdcb9e51d6948de53cbf556e1e4caa385907472a277a51550b1f5","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043079,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":4}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=4 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(4, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (4, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0080","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"236d643dcd36fa6f67684286995914200fb3a20b060701757d5aa4cd73e56af0","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043080,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":5}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=5 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(5, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (5, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0081","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37aa4dfc7da003129d297d748884edf7e31de5e7ab6512f340034d5da0ad08d3","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043081,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":6}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=6 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(6, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (6, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0082","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdcca46b1edf318d72f8bb9aa910798cd2f67413afcf382e4629576e85b074ef","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043082,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":7}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=7 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(7, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (7, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0083","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0026405ee0375f4dba74341b3a779bcf543cb3b9821897ff14329b5fe36002fd","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043083,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":8}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=8 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(8, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (8, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0084","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a661d93678148f1d12bf638f1daf1db1a133f30d16b34b00e3254cff7dabc9f","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043084,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":9}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=9 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(9, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (9, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0085","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c5055b3ab5d5fb4e03462c3ab1d954c3e5e276afe76931869303944d17bce95","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043085,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":10}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=10 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(10, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (10, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0086","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d7733464f612cd78c979690b4ab3afa9ecacc4c2c03ee0fae87fddf452b22a7","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043086,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":11}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=11 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(11, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (11, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0087","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81b3a71805cea2ca7726267480b1137731e98b074855bd6e3ae454a5c813d99f","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043087,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":12}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=12 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(12, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (12, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0088","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b39e9a00670c7aef65a2c5c4235e6fb0cea9aec5bb66e2ead924869d24fddaf2","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043088,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":13}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=13 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(13, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (13, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0089","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"087f204c502d572c18d8a11dc0aa2349c2de2dfce268f94952d03e1545808d7a","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043089,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":14}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=14 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(14, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (14, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0090","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0a6443ea8718fed871609e3e13d77c8742cd456dfb44c030c6cc66cfbdaf5b8","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043090,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":15}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=15 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(15, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (15, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0091","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"350e5a4ab4922b69a74638f121fecf8e75202ca0892ca8dd85f54b3b25f419b4","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043091,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":16}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=16 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(16, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (16, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0092","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d0054eac67f5673b192e289718d4511b1163f2244f6d633cde3e1f4d0ca5cbb","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043092,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":17}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=17 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(17, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (17, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0093","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e02dc09cc18fca74f3bf8716b946fb86eb502b27ac4772869cb9d2719f16f82b","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043093,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":18}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=18 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(18, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (18, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0094","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9fddc5a3905d41b4d0f4e09306b9f00d18040039a1186746e0320d6abef456be","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043094,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":19}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=19 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(19, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (19, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0095","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff4a33e5ea207a986c717c7d8c7c63cc54791396102a4a01d69f1a4a8c197ede","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043095,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":20}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=20 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(20, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (20, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0096","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be790238152132aa2979ca19745a9cc3579b2b25a76b83b4c7ae6df5e0a015fa","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043096,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":21}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=21 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(21, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (21, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0097","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c4ad783833fb5afcf3d7b2e78e9b4f91aec1e015df14c9281481557b88db4c6","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043097,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":22}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=22 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(22, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (22, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0098","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c17801878a3a840d78d9e7fceb86547e530bb3840d7f4cbbbc5bc1f701df082","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043098,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":23}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=23 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(23, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (23, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0099","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ed14e9f79135bd6baa4ce418c8aa29b727dde97babfd60c3ae77905d3417838","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043099,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":24}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=24 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(24, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (24, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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-20261104-r03-absolute-convergence-of-series-0100","batch_id":"batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8567e5e2bfa6563bfa455348fb21628d03448111e286254fd30305da012690c","family_id":"qf-mod35-absseries-term-zero-boundary","seed":202611043100,"semantic_template":"qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":25}"},"fields":{"title":"Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion","stem":"Use the shifted-harmonic boundary with positive shift=25 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).","answer":{"kind":"expression","value":"(25, 5, Yes, No, term_zero_not_absolute_convergence)","explanation":"The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (25, 5, Yes, No, term_zero_not_absolute_convergence)."},"skills":["Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion"],"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."}}