{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-series-comparison-test-01",
  "mko_id": "mko-series-comparison-test",
  "title": {
    "zh-Hant": "最終比較的方向與有限前綴",
    "en": "The direction of eventual comparison and finite prefixes"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-series-comparison-test-01",
    "source": {
      "mko_id": "mko-series-comparison-test",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "35951480a474b09d26963c54a0de77dde9fa5205f50b85ed9197df450d8c311b",
      "english_sha256": "5dc3868a6a23f13d44f069dc88247db8d5be1337575bf94194456850ec9be158"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "algebra",
      "epsilon",
      "nonnegative_series"
    ],
    "goal": {
      "zh-Hant": "證一個有任意有限前綴的級數由幾何尾上比較收斂，並示下比較發散。",
      "en": "Prove convergence using geometric upper comparison despite a finite prefix, and show divergent lower comparison."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "a_0=5,a_1=-1，a_n=(3/4)2^{-n}（n≥2）；b_n=2^{-n}（n≥2）。",
          "en": "a_0=5,a_1=-1; a_n=(3/4)*2^{-n} for n≥2; b_n=2^{-n} for n≥2."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "n\\ge2",
        "after_tex": "0\\le a_n=(3/4)2^{-n}\\le2^{-n}=b_n",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "係數 3/4 介於 0 與 1，證每個尾項的同方向比較。",
          "en": "The coefficient 3/4 lies between zero and one, proving the same-direction comparison for every tail term."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "未對負前綴 a_1 強加非負比較。",
            "en": "No nonnegative comparison is imposed on negative prefix term a_1."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S02",
        "uses": [
          "G1"
        ],
        "before_tex": "\\sum_{n=2}^{M}b_n",
        "after_tex": "\\sum_{n=2}^{M}2^{-n}=1/2-2^{-M}\\to1/2",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "有限幾何相減給上比較級數的獨立收斂證書。",
          "en": "Finite geometric subtraction independently certifies convergence of the upper reference series."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "參考級數收斂不是從被比較級數猜來。",
            "en": "Reference convergence is not guessed from the compared series."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S03",
        "uses": [
          "S01",
          "S02"
        ],
        "before_tex": "A_M=\\sum_{n=2}^{M}a_n",
        "after_tex": "0\\le A_M\\le\\sum_{n=2}^{M}b_n\\le1/2",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "逐項非負比較相加得到所有尾部分和上界。",
          "en": "Summing the nonnegative termwise bounds gives an upper bound on every tail partial sum."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "不等式涵蓋所有 M≥2，不是有限 grid。",
            "en": "The inequality covers every M≥2, not a finite grid."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S04",
        "uses": [
          "S01",
          "S03"
        ],
        "before_tex": "A_M\\text{ nondecreasing and bounded above}",
        "after_tex": "\\sum_{n=2}^{\\infty}a_n\\text{ converges}",
        "relation": "theorem_application",
        "rule": "nonnegative_series",
        "conditions": [],
        "explanation": {
          "zh-Hant": "非負性給單調，上界給完備性可用，這就是正向比較的實際證法。",
          "en": "Nonnegativity gives monotonicity and the bound enables completeness, which is the actual upper-comparison proof."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "若允許任意負尾項，這個單調證法失效。",
            "en": "Arbitrary negative tail terms would invalidate this monotonicity proof."
          }
        },
        "leaf_basis": "nonnegative_series"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S02",
          "S04"
        ],
        "before_tex": "\\sum_{n=0}^{M}a_n=5-1+(3/4)\\sum_{n=2}^{M}2^{-n}",
        "after_tex": "\\sum_{n=0}^{\\infty}a_n=4+3/8=35/8",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整公式還能算精確和；有限前綴只加 4。",
          "en": "The full law also yields the exact sum; the finite prefix adds only four."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "不把前綴符號當影響收斂性的尾證書。",
            "en": "Prefix signs are not treated as a tail convergence certificate."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S06",
        "uses": [
          "G1"
        ],
        "before_tex": "c_n=1,\\quad d_n=1+2^{-n}\\quad(n\\ge2)",
        "after_tex": "0\\le c_n\\le d_n,\\quad\\sum_{n=2}^{M}c_n=M-1",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "另取完整非負下比較，常數項的和無界。",
          "en": "Use a separate complete nonnegative lower comparison; the constant-term sums are unbounded."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "新 c,d 與主 a,b 已清楚區分。",
            "en": "The new c,d laws are distinct from the main a,b laws."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S07",
        "uses": [
          "S06"
        ],
        "before_tex": "\\sum_{n=2}^{M}d_n\\ge M-1\\to\\infty",
        "after_tex": "\\sum d_n\\text{ diverges}",
        "relation": "implies",
        "rule": "nonnegative_series",
        "conditions": [],
        "explanation": {
          "zh-Hant": "發散的是下界級數才可推較大非負級數發散。",
          "en": "Divergence of the lower reference forces divergence of the larger nonnegative series."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S06"
          ],
          "explanation": {
            "zh-Hant": "發散上界沒有同方向結論。",
            "en": "A divergent upper bound would give no such conclusion."
          }
        },
        "leaf_basis": "nonnegative_series"
      },
      {
        "id": "S08",
        "uses": [
          "S01",
          "S04",
          "S06",
          "S07"
        ],
        "before_tex": "0\\le u_n\\le v_n\\text{ eventually}",
        "after_tex": "\\sum v_n\\text{ convergent}\\Rightarrow\\sum u_n\\text{ convergent};\\quad\\sum u_n\\text{ divergent}\\Rightarrow\\sum v_n\\text{ divergent}",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [
          "\\exists N\\ \\forall n\\ge N:\\ 0\\le u_n\\le v_n"
        ],
        "explanation": {
          "zh-Hant": "前面兩種部分和界論證對任意給定全尾比較有效，有限多前項由固定常數排除。",
          "en": "The two partial-sum arguments apply to any supplied whole-tail comparison; finitely many initial terms are removed as a fixed constant."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S04",
            "S06",
            "S07"
          ],
          "explanation": {
            "zh-Hant": "這是有前提的方向命題，不是由幾何實例外推未驗比較。",
            "en": "This is a conditional directional argument, not an extrapolation to unverified comparisons."
          }
        },
        "leaf_basis": "algebra"
      }
    ],
    "result": {
      "zh-Hant": "主級數收斂為 35/8；正下界 c_n=1 迫使 d 級數發散。",
      "en": "The main series converges to 35/8; the positive lower bound c_n=1 forces the d series to diverge."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "上界級數發散則較小級數必發散。",
          "en": "A divergent upper series forces a smaller series to diverge."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "2^{-n}≤1/n（n≥1）但前者收斂後者發散。",
          "en": "2^{-n}≤1/n for n≥1, but the former converges and the latter diverges."
        }
      },
      {
        "claim": {
          "zh-Hant": "有限多個比較成立就夠。",
          "en": "Finitely many comparisons suffice."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "需要每個 n≥N 的全稱尾端。",
          "en": "Every n≥N is required."
        }
      },
      {
        "claim": {
          "zh-Hant": "a_1=-1 使最終比較不能用。",
          "en": "a_1=-1 invalidates eventual comparison."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "前綴可分離，N=2 後全尾非負。",
          "en": "The prefix is separated; the entire tail after N=2 is nonnegative."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "對足夠大n明示0≤a_n≤b_n。若sum b_n收斂，則sum a_n收斂；若sum a_n發散，則sum b_n發散。題族以exact geometric tails、有限前綴與正下界證書核對；有限多項不影響級數收斂性。負項、反向不等式、只有有限多個sample或拿divergent upper bound推被比較級數發散，都不滿足判別責任。",
          "en": "For every sufficiently large n require0≤a_n≤b_n. Convergence of the b series implies convergence of the a series; divergence of the a series implies divergence of the b series. Exact families use geometric tails, arbitrary finite prefixes and positive constant lower bounds. Finitely many terms do not change convergence, but every tail comparison is a universal obligation. Negative terms, a reversed inequality, finitely sampled comparisons or a divergent upper bound do not supply the required test."
        },
        "source": {
          "mko_id": "mko-series-comparison-test",
          "sha256": "35951480a474b09d26963c54a0de77dde9fa5205f50b85ed9197df450d8c311b",
          "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": "nonnegative_series",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "非負項部分和單調；若其有上界則由實數完備性收斂，若其無界則發散；有限前綴只加固定常數。",
          "en": "Nonnegative partial sums increase; if bounded they converge by real completeness, and if unbounded they diverge; a finite prefix only adds a fixed constant."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "sum-value",
        "lhs": "5-1+(3/4)*(1/2)",
        "rhs": "35/8",
        "step": "S05"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "明示 cutoff N=2 之後的全部非負項；有限前綴不限制符號。",
      "en": "Every nonnegative tail term after cutoff N=2 is explicit; finite-prefix signs are unrestricted."
    },
    "md_symbols": [
      [
        "N=2",
        "全尾比較起點",
        "whole-tail cutoff"
      ],
      [
        "A_M",
        "非負尾部分和",
        "nonnegative-tail partial sum"
      ]
    ],
    "md_error_controls": [
      {
        "title": "方向 / Direction",
        "zh": "收斂看上界，發散看下界。",
        "en": "Use an upper reference for convergence and a lower reference for divergence."
      },
      {
        "title": "最終量詞 / Eventual quantifier",
        "zh": "指定 N 後對所有後項證非負比較。",
        "en": "After specifying N prove nonnegative comparison for every later term."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "collection": "coverage-20261021",
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    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
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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."
  }
}