{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261117-r02-ratio-test-series-0020",
  "pilot_number": "P250-0195",
  "number": 195,
  "family_id": "qf-mod48-ratio-nonconstant-convergent",
  "title": {
    "zh-Hant": "第 195 題：級數比值判別法",
    "en": "Question 195: A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261117-r02-ratio-test-series-0020",
    "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-nonconstant-convergent；資料={\"scale\":20,\"ratio\":[5,11],\"shift\":20,\"alternating\":true}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(20, 5/11, 20, 是, a_n=20*(-1)^n*(5/11)^n/(n+20), abs_ratio=5/11*(n+20)/(n+21), limit=5/11, 8/11, 是, eventual_geometric_domination, absolutely_convergent)",
      "explanation_zh": "Wave6第十一層核對=(20, 5/11, 20, 是, a_n=20*(-1)^n*(5/11)^n/(n+20), abs_ratio=5/11*(n+20)/(n+21), limit=5/11, 8/11, 是, eventual_geometric_domination, absolutely_convergent)；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": 3,
      "skills": [
        "級數比值判別法"
      ]
    },
    "variant": {
      "family_id": "qf-mod48-ratio-nonconstant-convergent",
      "seed": 202611172020,
      "parameters": {
        "scale": 20,
        "ratio": [
          5,
          11
        ],
        "shift": 20,
        "alternating": true,
        "domain": "Exact_ratio_test_series",
        "operation": "ratio-nonconstant-convergent",
        "semantic_template": "ratio-nonconstant-convergent|{alternating:true,ratio:[5,11],scale:[20,1],shift:20}"
      }
    },
    "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": {
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    "question_id": "q-20261015-advance-20261117-r02-ratio-test-series-0020",
    "batch_id": "batch-20261015-advance-20261117-r02-ratio-test-series-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "836ee0d37df3e69075d51c6cb813d487448f332074bda1816eaa94c6f8a13ccd",
      "family_id": "qf-mod48-ratio-nonconstant-convergent",
      "seed": 202611172020,
      "semantic_template": "ratio-nonconstant-convergent|{alternating:true,ratio:[5,11],scale:[20,1],shift:20}"
    },
    "fields": {
      "title": "A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit",
      "stem": "Use exact scales A=20,B=20, rational ratio r=5/11. Set A=20>0,r=5/11∈(0,1), shift=20, alternating=true. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion).",
      "answer": {
        "kind": "expression",
        "value": "(20, 5/11, 20, Yes, a_n=20*(-1)^n*(5/11)^n/(n+20), abs_ratio=5/11*(n+20)/(n+21), limit=5/11, 8/11, Yes, eventual_geometric_domination, absolutely_convergent)",
        "explanation": "The entire rational base is exponentiated. The absolute ratio is r*(n+shift)/(n+shift+1), whose limit is r. Choose q=(r+1)/2 between r and1 for geometric domination and absolute convergence. The current answer exponentiates the entire parenthesized rational base, not p/(q^n); historical ambiguous displays remain in the retained original batch. The exact result, retaining the requested order, is (20, 5/11, 20, Yes, a_n=20*(-1)^n*(5/11)^n/(n+20), abs_ratio=5/11*(n+20)/(n+21), limit=5/11, 8/11, Yes, eventual_geometric_domination, absolutely_convergent)."
      },
      "skills": [
        "A Nonconstant Absolute Ratio Is Controlled below One by a Declared Rational Limit"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "eventually nonzero、absolute successive ratio與limit存在責任。操作=ratio-nonconstant-convergent；資料={\"scale\":20,\"ratio\":[5,11],\"shift\":20,\"alternating\":true}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use exact scales A=20,B=20, rational ratio r=5/11. Set A=20>0,r=5/11∈(0,1), shift=20, alternating=true. The sequence is a_n=A*(r)^n/(n+shift), multiplied by (−1)^n exactly when alternating is true. Give (scale, ratio, shift, alternating, term_description, absolute_ratio_description, limit_description, dominating_ratio, ratio_bracket, domination_statement, conclusion)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "a_n=20*(-1)^n*(5/11)^n/(n+20)",
      "after": "|a_n|=20·(5/11)^n/(n+20)>0 for every integer n≥0",
      "relation": "implies",
      "rule_id": "absolute_value",
      "conditions": [
        "n≥0",
        "A=20>0",
        "0<r=5/11<1",
        "s=20≥1"
      ],
      "explanation": {
        "zh-Hant": "整個有理數底數都被 n 次方；若有交錯符號，其絕對值為 1。",
        "en": "The whole rational base is raised to n; an alternating sign, if present, has absolute value 1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "正的 A、r 與 n+s 保證後面每次比值的分母都不為零。",
          "en": "Positive A, r and n+s ensure every following ratio denominator is nonzero."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "|a_n|=20·(5/11)^n/(n+20)",
      "after": "|a_(n+1)|=20·(5/11)^(n+1)/(n+21)",
      "relation": "implies",
      "rule_id": "index_shift",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個出現 n 的位置都加 1，包括分母的平移。",
        "en": "Replace every occurrence of n by n+1, including the shifted denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "不可只改指數而保留原分母。",
          "en": "Do not shift the exponent while leaving the denominator unchanged."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "|a_(n+1)| / |a_n|",
      "after": "abs_ratio=5/11*(n+20)/(n+21)",
      "relation": "implies",
      "rule_id": "positive_factor_cancellation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把除法改乘倒數，約去 A 與 r^n；前一步已證所約掉的量皆非零。",
        "en": "Multiply by the reciprocal and cancel A and r^n; their nonzero status was established above."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "保留絕對比值；不是用可能為負的有號相鄰比。",
          "en": "Keep the absolute ratio, not a potentially negative signed ratio."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "abs_ratio=5/11*(n+20)/(n+21)",
      "after": "R_n=5/11−(5/11)/(n+21); |R_n−r|=(5/11)/(n+21)",
      "relation": "implies",
      "rule_id": "rational_error_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將分子寫成分母減 1，就把極限問題化成明確的正誤差。",
        "en": "Write the numerator as the denominator minus 1 to obtain the explicit positive error."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "趨零的是誤差，不是分母；分母隨 n 增大。",
          "en": "The error tends to zero while the denominator grows."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "epsilon>0; N=floor(r/epsilon)+1",
      "after": "n≥N ⇒ n+21>r/epsilon ⇒ |R_n−r|<epsilon",
      "relation": "implies",
      "rule_id": "archimedean_tail",
      "conditions": [
        "epsilon>0"
      ],
      "explanation": {
        "zh-Hant": "由實數 Archimedean 性，這個整數 N 存在；正數取倒數反向，所以控制的是全部尾端而非有限樣本。",
        "en": "The Archimedean property supplies this integer N. Taking reciprocals of positive numbers reverses the inequality and controls the entire tail, not a finite sample."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "N 只依 epsilon 與固定參數，不依之後選的 n。",
          "en": "N depends only on epsilon and the fixed parameters, not on a later n."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "q−r := sub(8/11, 5/11)",
      "after": "q−r = 3/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "取 q=(r+1)/2；先核對 q 嚴格大於 r。",
        "en": "Set q=(r+1)/2 and check that q is strictly greater than r."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 q−r，不以小數近似替代。",
          "en": "Recompute q−r with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "8/11",
          "5/11"
        ],
        "output": "3/11",
        "label": "q−r"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "1−q := sub(1, 8/11)",
      "after": "1−q = 3/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再核對 q 嚴格小於 1。",
        "en": "Then check that q is strictly below 1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 1−q，不以小數近似替代。",
          "en": "Recompute 1−q with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "8/11"
        ],
        "output": "3/11",
        "label": "1−q"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "|a_0| := div(20, 20)",
      "after": "|a_0| = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "n=0 時底數的零次方為 1，因此求出幾何控制的起始常數。",
        "en": "At n=0 the positive base to the zeroth power is 1; compute the initial domination constant."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 |a_0|，不以小數近似替代。",
          "en": "Recompute |a_0| with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "20",
          "20"
        ],
        "output": "1",
        "label": "|a_0|"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "0<R_n<r=5/11<q=8/11<1",
      "after": "|a_n|≤(1)·(8/11)^n for every n≥0",
      "relation": "implies",
      "rule_id": "inductive_geometric_domination",
      "conditions": [],
      "explanation": {
        "zh-Hant": "n=0 時為等號；若第 n 項受控，乘上 R_n<q 即控制第 n+1 項。由歸納法取得全部 n 的界。",
        "en": "Equality holds at n=0. If the bound holds at n, multiplying by R_n<q proves it at n+1. Induction gives the bound for every n."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "此為明示的歸納推論，不把檢查若干比值當成無限尾端證明。",
          "en": "This is an explicit induction argument, not an inference from finitely many checked ratios."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "sum from k=0 to M of |a_k| ≤ (1)·(1−(8/11)^(M+1))/(1−(8/11))",
      "after": "all absolute partial sums ≤ 11/3",
      "relation": "implies",
      "rule_id": "finite_geometric_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "因 0<q<1，有限幾何和有此上界；絕對值部分和又單調遞增，依實數完備性收斂。",
        "en": "Since 0<q<1 the finite geometric sum has this bound. Absolute partial sums are increasing and bounded, so completeness of the reals gives convergence."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "這裡使用單調有界收斂與幾何和作為明示前提，不宣稱 Lean 已證。",
          "en": "Monotone bounded convergence and the geometric-sum identity are explicit prerequisites, not a claim of a Lean proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "M>N≥0",
      "after": "sum from k=N+1 to M of |a_k| ≤ (1)·(8/11)^(N+1)/(1−(8/11)) → 0",
      "relation": "implies",
      "rule_id": "tail_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一幾何界還給出任意兩個尾端部分和之差的上界，說明為什麼整個級數絕對收斂。",
        "en": "The same geometric bound controls differences of tail partial sums, explaining absolute convergence of the entire series."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "結論對有無交錯符號均成立，但不求本題未要求的級數精確和。",
          "en": "The conclusion holds with or without alternating signs; no unrequested exact series sum is claimed."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "A": "20",
    "r": "5/11",
    "s": 20,
    "alternating": true,
    "q": "8/11",
    "first_absolute": "1",
    "geometric_total_bound": "11/3"
  },
  "conclusion": {
    "canonical_value": "(20, 5/11, 20, 是, a_n=20*(-1)^n*(5/11)^n/(n+20), abs_ratio=5/11*(n+20)/(n+21), limit=5/11, 8/11, 是, eventual_geometric_domination, absolutely_convergent)",
    "explanation": {
      "zh-Hant": "此級數絕對收斂，且已給出共同尾端的幾何控制。",
      "en": "The series converges absolutely, with an explicit geometric tail bound."
    }
  },
  "definition_note": {
    "zh-Hant": "A>0、0<r<1、s≥1 且 n≥0；明示的誤差界、歸納與幾何界支持整個尾端。",
    "en": "Require A>0, 0<r<1, s≥1 and n≥0. The explicit error bound, induction and geometric bound apply to the entire tail."
  },
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