{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-ratio-test-series-01",
  "mko_id": "mko-ratio-test-series",
  "title": {
    "zh-Hant": "比值極限、幾何支配與 L=1",
    "en": "Ratio limits, geometric domination and L=1"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-ratio-test-series-01",
    "source": {
      "mko_id": "mko-ratio-test-series",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "90d6b9fa7fcadbd04ee6ef7efff16b13647e3de4bf46c8c2248ca8168ee49382",
      "english_sha256": "4cedc70d3305cd8238c8d23bc1dca04541d313107ee78217fb5a79d376d7ff3e"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "algebra",
      "epsilon",
      "real_complete"
    ],
    "goal": {
      "zh-Hant": "以完整有理比值證 L<1 收斂及 L>1 發散，並核 L=1 無結論。",
      "en": "Use complete rational ratios for L<1 convergence and L>1 divergence, then audit indecision at L=1."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "a_n=(1/2)^n/(n+1), b_n=2^n/(n+1)，全部 n≥0；另 c_n=1/(n+1)、d_n=1/(n+1)^2。",
          "en": "a_n=(1/2)^n/(n+1), b_n=2^n/(n+1) for every n≥0; also c_n=1/(n+1), d_n=1/(n+1)^2."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "a_n>0\\quad(n\\ge0)",
        "after_tex": "\\left|a_{n+1}/a_n\\right|=\\tfrac12\\frac{n+1}{n+2}=\\tfrac12-\\frac1{2(n+2)}",
        "relation": "equivalent",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "所有項非零，完整絕對比值可用精確式直接取極限。",
          "en": "All terms are nonzero, and the full absolute-ratio identity directly supplies its limit."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "未忽略絕對值，也不在零分母上計算。",
            "en": "Absolute values are retained and no zero denominator is used."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S02",
        "uses": [
          "S01"
        ],
        "before_tex": "\\frac1{2(n+2)}\\to0",
        "after_tex": "L_a=1/2<q=3/4<1",
        "relation": "implies",
        "rule": "epsilon",
        "conditions": [],
        "explanation": {
          "zh-Hant": "誤差小於 epsilon 可取 n>1/(2epsilon)，給真正極限及明示中間 q。",
          "en": "Taking n>1/(2epsilon) controls the error, proving the actual limit and an explicit intermediate q."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01"
          ],
          "explanation": {
            "zh-Hant": "L<q<1 三個關係皆嚴格。",
            "en": "All comparisons in L<q<1 are strict."
          }
        },
        "leaf_basis": "epsilon"
      },
      {
        "id": "S03",
        "uses": [
          "S01",
          "S02"
        ],
        "before_tex": "a_{n+1}/a_n<1/2<3/4\\quad(n\\ge0)",
        "after_tex": "0<a_n\\le(3/4)^n a_0=(3/4)^n",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "從 a_0=1 歸納全尾幾何支配。",
          "en": "Induct from a_0=1 to obtain geometric domination of the entire tail."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "支配是每個 n，不是僅一個小比值。",
            "en": "Domination covers every n, not merely one favorable ratio."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S04",
        "uses": [
          "S03",
          "real_complete"
        ],
        "before_tex": "\\sum_{n=0}^{N}a_n\\le\\sum_{n=0}^{N}(3/4)^n\\le4",
        "after_tex": "\\sum a_n\\text{ converges absolutely}",
        "relation": "implies",
        "rule": "real_complete",
        "conditions": [
          "\\text{bounded monotone real sequences converge}"
        ],
        "explanation": {
          "zh-Hant": "非負部分和單調且有界，由實數完備性得收斂；絕對值相同。",
          "en": "The nonnegative partial sums increase and are bounded; real completeness gives convergence, which is absolute here."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "real_complete"
          ],
          "explanation": {
            "zh-Hant": "需有界單調實數收斂性，並由幾何和獨立給上界。",
            "en": "Bounded-monotone real convergence is assumed, with the upper bound independently supplied by a geometric sum."
          }
        },
        "leaf_basis": "real_complete"
      },
      {
        "id": "S05",
        "uses": [
          "G1"
        ],
        "before_tex": "\\left|b_{n+1}/b_n\\right|=2\\frac{n+1}{n+2}=2-\\frac2{n+2}",
        "after_tex": "L_b=2>1,\\quad b_{n+1}/b_n\\ge3/2\\ (n\\ge2)",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整差式證極限 2，且 n≥2 時比值至少 3/2。",
          "en": "The full difference formula proves limit two, and for n≥2 the ratio is at least 3/2."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "大於 1 的界是整個尾端。",
            "en": "The greater-than-one bound holds on the entire tail."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S06",
        "uses": [
          "S05"
        ],
        "before_tex": "b_n\\ge b_2(3/2)^{n-2}\\quad(n\\ge2)",
        "after_tex": "b_n\\not\\to0;\\quad\\sum b_n\\text{ diverges}",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "歸納幾何增長，項甚至無界，所以違反級數收斂必要條件。",
          "en": "Induction gives geometric growth and unbounded terms, violating the necessary term condition for series convergence."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S05"
          ],
          "explanation": {
            "zh-Hant": "不是將 ratio>1 的一項單獨當發散證明。",
            "en": "A single ratio above one is not treated as a divergence proof."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S07",
        "uses": [
          "G1"
        ],
        "before_tex": "c_{n+1}/c_n=\\frac{n+1}{n+2},\\quad d_{n+1}/d_n=\\left(\\frac{n+1}{n+2}\\right)^2",
        "after_tex": "L_c=L_d=1",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩個有理比值相對於 1 的誤差趨零，因此極限同為 1；不是分母本身趨零，而且兩個比值都不是常數。",
          "en": "The errors of the two rational ratios relative to 1 tend to zero, giving the same limit 1; their denominators do not tend to zero, and neither ratio is constant."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "沒有把 L=1 偷換為每一步 ratio=1。",
            "en": "A limit of one is not confused with every ratio equaling one."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07"
        ],
        "before_tex": "\\sum c_n\\text{ diverges by dyadic blocks};\\quad\\frac1{(n+1)^2}\\le\\frac1{n(n+1)}\\ (n\\ge1)",
        "after_tex": "\\sum d_n\\text{ converges by telescoping upper comparison};\\quad L=1\\text{ is inconclusive}",
        "relation": "implies",
        "rule": "epsilon",
        "conditions": [],
        "explanation": {
          "zh-Hant": "調和每個二進位塊≥1/2；平方倒數尾上界望遠鏡和 1，因此同 L 有相反結果。",
          "en": "Each harmonic dyadic block is at least 1/2; reciprocal-square tails are bounded above by a telescoping series of sum one, giving opposite statuses at the same L."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07"
          ],
          "explanation": {
            "zh-Hant": "判別無結論並不代表原級數沒有收斂性狀態。",
            "en": "An inconclusive test does not mean the series has no convergence status."
          }
        },
        "leaf_basis": "epsilon"
      }
    ],
    "result": {
      "zh-Hant": "a 絕對收斂、b 發散；調和與平方倒數在 L=1 給相反結果。",
      "en": "a converges absolutely, b diverges, and harmonic versus reciprocal-square series give opposite outcomes at L=1."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "L=1 代表收斂。",
          "en": "L=1 means convergence."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "c 是發散調和級數。",
          "en": "c is divergent harmonic."
        }
      },
      {
        "claim": {
          "zh-Hant": "只看前十個比值小於 1 就證收斂。",
          "en": "Ten ratios below one prove convergence."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "後續尾端可能變化，需極限或全尾支配。",
          "en": "The later tail may differ; a limit or whole-tail domination is required."
        }
      },
      {
        "claim": {
          "zh-Hant": "有零項仍可無條件除相鄰項。",
          "en": "Zero terms permit unconditional adjacent division."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "分母為零時比值未定，需另處理。",
          "en": "A zero denominator makes the ratio undefined and requires separate treatment."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "對eventually nonzero序列明示L=lim |a_(n+1)/a_n|。L<1時另取L<q<1並建立eventual geometric domination而得absolute convergence；L>1時項絕對值最終不趨零而發散；L=1時判別無結論，p-series的p=2與p=1提供相反結果。必須取absolute ratio、證明極限而非只算有限多項，zero terms與不存在的ratio limit另行處理。",
          "en": "Declare the existing limit L=lim|a_(n+1)/a_n| on an eventually nonzero tail. When L<1 choose L<q<1 and establish eventual geometric domination to obtain absolute convergence. When L>1 the term magnitudes eventually fail to tend to 0, so the series diverges. L=1 gives no decision: reciprocal-square and harmonic tails have opposite statuses with that same limit. Use absolute ratios and an actual limit argument, not a finite favorable prefix. Zero denominators and missing limits are separate boundary cases."
        },
        "source": {
          "mko_id": "mko-ratio-test-series",
          "sha256": "90d6b9fa7fcadbd04ee6ef7efff16b13647e3de4bf46c8c2248ca8168ee49382",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "algebra",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "實數域的等式運算；除法前分母必非零，不等式乘正數保持方向。",
          "en": "Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities."
        }
      },
      {
        "id": "epsilon",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "收斂表示每個正 epsilon 有一個整數 N，使全部 n≥N 的誤差嚴格小於 epsilon；使用實數 Archimedean 性。",
          "en": "Convergence requires for every positive epsilon an integer N controlling every n≥N by a strict error bound; use the Archimedean property of the reals."
        }
      },
      {
        "id": "real_complete",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "非空有上界的實集合有 supremum，故單調有界實數列收斂；這也適用非負級數有界部分和。",
          "en": "Every nonempty bounded-above real set has a supremum, so bounded monotone real sequences converge, including bounded partial sums of nonnegative series."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "growth-at-two",
        "lhs": "2*(2+1)/(2+2)",
        "rhs": "3/2",
        "step": "S05"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "先驗尾端非零與絕對比值極限，不從有限 ratios 推極限。",
      "en": "Qualify eventual nonzero terms and absolute ratio limits, not limits inferred from finite ratios."
    },
    "md_symbols": [
      [
        "L",
        "已證存在的絕對比值極限",
        "proved absolute-ratio limit"
      ],
      [
        "q",
        "L 與 1 間支配比",
        "domination ratio between L and one"
      ]
    ],
    "md_error_controls": [
      {
        "title": "絕對比 / Absolute ratios",
        "zh": "先核非零尾端，再對絕對比值證極限。",
        "en": "Qualify a nonzero tail before proving its absolute-ratio limit."
      },
      {
        "title": "臨界值 / Critical value",
        "zh": "L=1 用其他獨立證書，不硬給結論。",
        "en": "At L=1 use independent arguments instead of forcing a decision."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
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    "glossary_reader_paraphrases": [
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    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}