{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261104-r03-absolute-convergence-of-series-0017",
  "pilot_number": "R0034-17967",
  "number": 17967,
  "family_id": "qf-mod35-absseries-geometric",
  "title": {
    "zh-Hant": "第 17967 題：絕對收斂",
    "en": "Question 17967: Sum Absolute Terms, Not the Absolute Value of the Signed Sum"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261104-r03-absolute-convergence-of-series-0017",
    "batch_id": "batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-absolute-convergence-of-series"
    ],
    "title_zh": "絕對收斂",
    "stem_zh": "partial sums與absolute-series exact convergence certificate。操作=absseries-geometric；資料={\"a\":[17,18],\"r\":[-1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(17/18, -1/2, 17/9, 是, absolute_geometric_certificate)",
      "explanation_zh": "Wave5第四層核對=(17/18, -1/2, 17/9, 是, absolute_geometric_certificate)；依完整明示資料重算；absolute convergence核sum absolute values並推出原級數收斂；term趨零 bounded或alternation不足；conditional需雙證書。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "絕對收斂"
      ]
    },
    "variant": {
      "family_id": "qf-mod35-absseries-geometric",
      "seed": 202611043017,
      "parameters": {
        "a": [
          17,
          18
        ],
        "r": [
          -1,
          2
        ],
        "domain": "Exact_absolute_or_conditional_series",
        "operation": "absseries-geometric",
        "semantic_template": "qf-mod35-absseries-geometric/{\"a\":[17,18],\"r\":[-1,2]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "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."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "partial sums與absolute-series exact convergence certificate。操作=absseries-geometric；資料={\"a\":[17,18],\"r\":[-1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "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)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=absseries-geometric;original question snapshot retained;source presentation interface resolved in this page",
      "after": "EXPLICIT ordered output fields=[a, r, absolute_series_sum, absolutely_convergent, certificate]",
      "relation": "implies",
      "rule_id": "absseries_explicit_source_interface_fields_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題快照保留歷史介面標記，這裡直接列完整欄位，不需瀏覽器JavaScript或外部查表。欄位名只是presentation metadata，不是數學運算；term-zero的第四個否是『條件是否充分』，不是誤標成另一個converges旗標。",
        "en": "Preserve the original snapshot's historical interface marker while listing the full ordered fields here,without browser JavaScript or an external lookup.Field labels are presentation metadata,not mathematical evaluation.The fourth term-zero No marks insufficiency,not a mislabeled second convergence flag."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "absolute geometric series one minus effective ratio := sub(1, 1/2)",
      "after": "absolute geometric series one minus effective ratio = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 absolute geometric series one minus effective ratio，不以小數近似替代。",
          "en": "Recompute absolute geometric series one minus effective ratio with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "1/2"
        ],
        "output": "1/2",
        "label": "absolute geometric series one minus effective ratio"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "absolute geometric series exact infinite candidate sum := div(17/18, 1/2)",
      "after": "absolute geometric series exact infinite candidate sum = 17/9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 absolute geometric series exact infinite candidate sum，不以小數近似替代。",
          "en": "Recompute absolute geometric series exact infinite candidate sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "17/18",
          "1/2"
        ],
        "output": "17/9",
        "label": "absolute geometric series exact infinite candidate sum"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "absolute geometric series:original SIGNED a=17/18;SIGNED r=-1/2;absolute ratio=1/2<1;effective terms=(17/18)*(1/2)^k,k=0,1,...",
      "after": "candidate sum=17/9;effective denominator1-ratio=1/2 is nonzero",
      "relation": "implies",
      "rule_id": "series_geometric_exact_input_roles_and_absolute_ratio_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先保留原a、r的符號，依本題是否求絕對值級數才換有效係數及ratio；|r|<1是這份證書域，不用有限樣本或另種零係數例外改寫題目。",
        "en": "Retain signed a and r.Change to effective absolute values only when this operation asks for the absolute series.The |r|<1 guard is this certificate's domain;finite samples or an unrelated zero-coefficient exception do not change it."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "absolute geometric series:for ANY N>=1,the first N terms start at k0 with leading=17/18,ratio=1/2",
      "after": "(1-ratio)*S_N=(17/18)*(1-(1/2)^N);S_N=(17/18)*(1-(1/2)^N)/(1/2);candidate minus S_N=(17/18)*(1/2)^N/(1/2)",
      "relation": "implies",
      "rule_id": "series_geometric_all_N_finite_cancellation_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把有限和乘(1−ratio)，相鄰冪次抵消後只留首尾；這個恆等式對任意N≥1，並不是看幾個partial sums猜公式。",
        "en": "Multiply the finite sum by1−ratio and cancel adjacent powers,leaving endpoints.This identity holds for arbitrary N≥1,not a formula guessed from several partial sums."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "absolute geometric series:|effective ratio|=1/2<1;remainder magnitude=abs(17/18)*abs(1/2)^N/abs(1/2)",
      "after": "geometric power tends to0;remainder tends to0;partial sums converge to the exact candidate",
      "relation": "implies",
      "rule_id": "series_geometric_remainder_limit_not_finite_prefix_inference",
      "conditions": [],
      "explanation": {
        "zh-Hant": "嚴格小於1的幾何冪趨零，固定有限係數與非零分母保餘項趨零，因此partial-sum sequence才有此極限。",
        "en": "Powers of a ratio with absolute value strictly below1 tend to0.Fixed finite coefficients and a nonzero denominator preserve this remainder limit,which proves convergence of the partial-sum sequence."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "original signed a=17/18;r=-1/2;for EVERY k>=0,abs(a*r^k)=abs(a)*abs(r)^k;absolute-series sum=17/9",
      "after": "for every m>n>=1,absolute signed tail<=absolute-series tail=abs(a)*abs(r)^n/(1-abs(r)) tending to0;Cauchy criterion and real completeness give convergence of original series",
      "relation": "implies",
      "rule_id": "absseries_absolute_convergence_implies_original_via_cauchy_tail",
      "conditions": [],
      "explanation": {
        "zh-Hant": "絕對值作用在每一項，得到|a|/(1−|r|)，不是|a/(1−r)|。任意有號尾和由絕對尾和控制至0，Cauchy與實數完備性才給原級數收斂，並非有限prefix外推。",
        "en": "Absolute values act termwise,giving |a|/(1−|r|),not |a/(1−r)|.Every signed tail is bounded by an absolute tail tending to0;the Cauchy criterion and real completeness then give original-series convergence,not extrapolation from a finite prefix."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "17/18",
      "-1/2",
      "17/9",
      true,
      "absolute_geometric_certificate"
    ]
  },
  "conclusion": {
    "canonical_value": "(17/18, -1/2, 17/9, 是, absolute_geometric_certificate)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
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        "original_path": "public/data/mko/absolute-convergence-of-series.json",
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    "selected_part_sha256": "d5b0037cbaeef26e3828d04bf29e0e0f6e466926098b6b09e3237d1fdbdbd0e2",
    "original_candidate_sha256": "6375b4f5c7606fc2aac9cada73500f0673103380ede5f986be28fedecc35b1cc",
    "original_candidate_line_sha256": "6375b4f5c7606fc2aac9cada73500f0673103380ede5f986be28fedecc35b1cc"
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  "historical_source_state": {
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      "source_question_state": "candidate",
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      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
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    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-008-p35",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
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    "bilingual_semantics": true,
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    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
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    "formalization": "not_formalized",
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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}