{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261117-r02-ratio-test-series-0026",
  "pilot_number": "P1000-0701",
  "number": 701,
  "family_id": "qf-mod48-ratio-geometric-divergent",
  "title": {
    "zh-Hant": "第 701 題：級數比值判別法",
    "en": "Question 701: An Absolute Geometric Ratio above One Violates the Necessary Term Limit"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261117-r02-ratio-test-series-0026",
    "batch_id": "batch-20261015-advance-20261117-r02-ratio-test-series-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-ratio-test-series"
    ],
    "title_zh": "級數比值判別法",
    "stem_zh": "eventually nonzero、absolute successive ratio與limit存在責任。操作=ratio-geometric-divergent；資料={\"scale\":1,\"ratio\":[6,3],\"alternating\":false}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(1, 2, 否, 2, terms_do_not_tend_to_zero, divergent)",
      "explanation_zh": "Wave6第十一層核對=(1, 2, 否, 2, terms_do_not_tend_to_zero, divergent)；依完整明示資料獨立重算；L<1絕對收斂、L>1發散、L=1無結論；有限ratio samples、zero denominators或raw signed ratio不能冒充假設。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "級數比值判別法"
      ]
    },
    "variant": {
      "family_id": "qf-mod48-ratio-geometric-divergent",
      "seed": 202611172026,
      "parameters": {
        "scale": 1,
        "ratio": [
          6,
          3
        ],
        "alternating": false,
        "domain": "Exact_ratio_test_series",
        "operation": "ratio-geometric-divergent",
        "semantic_template": "ratio-geometric-divergent|{alternating:false,ratio:[2,1],scale:[1,1]}"
      }
    },
    "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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261117-r02-ratio-test-series-0026",
    "batch_id": "batch-20261015-advance-20261117-r02-ratio-test-series-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
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      "family_id": "qf-mod48-ratio-geometric-divergent",
      "seed": 202611172026,
      "semantic_template": "ratio-geometric-divergent|{alternating:false,ratio:[2,1],scale:[1,1]}"
    },
    "fields": {
      "title": "An Absolute Geometric Ratio above One Violates the Necessary Term Limit",
      "stem": "Use positive scale=1, exact ratio=6/3>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).",
      "answer": {
        "kind": "expression",
        "value": "(1, 2, No, 2, terms_do_not_tend_to_zero, divergent)",
        "explanation": "The absolute successive ratio is the declared ratio and term magnitudes do not tend to0, even with alternating signs. This proves divergence for the specified geometric instance, not a general theorem from finite samples. The exact result, retaining the requested order, is (1, 2, No, 2, terms_do_not_tend_to_zero, divergent)."
      },
      "skills": [
        "An Absolute Geometric Ratio above One Violates the Necessary Term Limit"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "eventually nonzero、absolute successive ratio與limit存在責任。操作=ratio-geometric-divergent；資料={\"scale\":1,\"ratio\":[6,3],\"alternating\":false}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use positive scale=1, exact ratio=6/3>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "A=1>0; r=2>1; alternating=否",
      "after": "|a_n|=1·(2)^n",
      "relation": "implies",
      "rule_id": "absolute_geometric_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "可能的交錯符號在絕對值中消失，底數的整體被乘方。",
        "en": "Any alternating sign disappears in absolute value, and the entire base is exponentiated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "有限核對不證無限尾端；本步的量詞、非零分母與絕對值條件仍須保留。",
          "en": "Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "|a_(n+1)|/|a_n|=(1·(2)^(n+1))/(1·(2)^n)",
      "after": "absolute ratio=2",
      "relation": "implies",
      "rule_id": "cancel_nonzero_factors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "A與r均為正，所以可約掉整個非零公因子。",
        "en": "A and r are positive, so the complete common nonzero factor may be cancelled."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "有限核對不證無限尾端；本步的量詞、非零分母與絕對值條件仍須保留。",
          "en": "Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "r=2>1",
      "after": "|a_n|≥A=1>0 for every n≥0",
      "relation": "implies",
      "rule_id": "nonzero_term_obstruction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "取epsilon=A/2，沒有任何尾端能使所有項的絕對值小於epsilon。",
        "en": "With epsilon=A/2, no tail can make every term magnitude smaller than epsilon."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "有限核對不證無限尾端；本步的量詞、非零分母與絕對值條件仍須保留。",
          "en": "Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "convergent series ⇒ its terms tend to0",
      "after": "these terms do not tend to0 ⇒ series diverges",
      "relation": "implies",
      "rule_id": "necessary_term_limit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "收斂部分和之差a_n必趨零；本題違反此必要條件。",
        "en": "If partial sums converged, their successive differences a_n would tend to0; this necessary condition fails."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "有限核對不證無限尾端；本步的量詞、非零分母與絕對值條件仍須保留。",
          "en": "Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "1",
      "2",
      false,
      "2",
      "terms_do_not_tend_to_zero",
      "divergent"
    ]
  },
  "conclusion": {
    "canonical_value": "(1, 2, 否, 2, terms_do_not_tend_to_zero, divergent)",
    "explanation": {
      "zh-Hant": "上述完整代值與條件核對得到原題答案；欄位次序及真假標記保持原始合約。",
      "en": "The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings."
    }
  },
  "definition_note": {
    "zh-Hant": "A>0、|r|>1 的幾何項；|a_n|≥A，項不趨於 0，故級數發散。",
    "en": "Geometric terms with A>0 and |r|>1 satisfy |a_n|≥A; the terms do not tend to 0, so the series diverges."
  },
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      "en": "Source-bound pedagogical derivation candidate: exact arithmetic and finite enumeration are not independent AI 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."
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