"Abel sum" meaning in English

{
  "etymology_text": "After Norwegian mathematician Niels Henrik Abel (1802-1829).",
  "forms": [
    {
      "form": "Abel sums",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Abel sum (plural Abel sums)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w"
        },
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          "name": "Pages with 1 entry",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with entries",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Mathematical analysis",
          "orig": "en:Mathematical analysis",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "derived": [
        {
          "word": "Abel summable"
        },
        {
          "word": "Abel summation"
        }
      ],
      "examples": [
        {
          "ref": "1967, Jan Mikusiński, Operational Calculus, Cambridge University Press, page 102, The Abel sum of ∑a_n is defined as the limit of the corresponding power series",
          "text": "lim _(x→1-0)∑ₙ₌₀ ᪲a_nxⁿ.\nThe existence of the Abel sum is ascertained when the series in question is known to be summable (C, r) for some value of r."
        },
        {
          "ref": "2005, Bulletin of the American Mathematical Society, page 81:",
          "text": "Jacobi in his Vorlesungen über Dynamik [1884] had used Abel sums to separate variables in the Hamilton-Jacobi equation in connection with the geodesic flow on the surface of a 3-dimensional ellipsoid, etc.",
          "type": "quote"
        },
        {
          "ref": "2012, Peter L. Duren, Invitation to Classical Analysis, American Mathematical Society, page 180:",
          "text": "Also, Abel's theorem guarantees that the Abel sum of a convergent series exists and is equal to the ordinary sum.",
          "type": "quote"
        }
      ],
      "glosses": [
        "Given a power series f(x)=∑ₙ₌₀ ᪲a_nxⁿ that is convergent for real x in the open interval (0, 1), the value lim _(x→1⁻)∑ₙ₌₀ ᪲a_nxⁿ, which is assigned to f(1)=∑ₙ₌₀ ᪲a_n by the Abel summation method (or A-method)."
      ],
      "id": "en-Abel_sum-en-noun-jU328QT7",
      "links": [
        [
          "mathematical analysis",
          "mathematical analysis"
        ],
        [
          "power series",
          "power series"
        ],
        [
          "convergent",
          "convergent"
        ],
        [
          "real",
          "real"
        ],
        [
          "open interval",
          "open interval"
        ],
        [
          "Abel summation method",
          "Abel summation method"
        ],
        [
          "A-method",
          "A-method"
        ]
      ],
      "raw_glosses": [
        "(mathematical analysis) Given a power series f(x)=∑ₙ₌₀ ᪲a_nxⁿ that is convergent for real x in the open interval (0, 1), the value lim _(x→1⁻)∑ₙ₌₀ ᪲a_nxⁿ, which is assigned to f(1)=∑ₙ₌₀ ᪲a_n by the Abel summation method (or A-method)."
      ],
      "related": [
        {
          "word": "Abel mean"
        },
        {
          "word": "Abel summation method"
        },
        {
          "word": "summability method"
        },
        {
          "word": "summation method"
        }
      ],
      "topics": [
        "mathematical-analysis",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "Niels Henrik Abel"
      ]
    }
  ],
  "word": "Abel sum"
}

{
  "derived": [
    {
      "word": "Abel summable"
    },
    {
      "word": "Abel summation"
    }
  ],
  "etymology_text": "After Norwegian mathematician Niels Henrik Abel (1802-1829).",
  "forms": [
    {
      "form": "Abel sums",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Abel sum (plural Abel sums)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "Abel mean"
    },
    {
      "word": "Abel summation method"
    },
    {
      "word": "summability method"
    },
    {
      "word": "summation method"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English eponyms",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "Pages with 1 entry",
        "Pages with entries",
        "en:Mathematical analysis"
      ],
      "examples": [
        {
          "ref": "1967, Jan Mikusiński, Operational Calculus, Cambridge University Press, page 102, The Abel sum of ∑a_n is defined as the limit of the corresponding power series",
          "text": "lim _(x→1-0)∑ₙ₌₀ ᪲a_nxⁿ.\nThe existence of the Abel sum is ascertained when the series in question is known to be summable (C, r) for some value of r."
        },
        {
          "ref": "2005, Bulletin of the American Mathematical Society, page 81:",
          "text": "Jacobi in his Vorlesungen über Dynamik [1884] had used Abel sums to separate variables in the Hamilton-Jacobi equation in connection with the geodesic flow on the surface of a 3-dimensional ellipsoid, etc.",
          "type": "quote"
        },
        {
          "ref": "2012, Peter L. Duren, Invitation to Classical Analysis, American Mathematical Society, page 180:",
          "text": "Also, Abel's theorem guarantees that the Abel sum of a convergent series exists and is equal to the ordinary sum.",
          "type": "quote"
        }
      ],
      "glosses": [
        "Given a power series f(x)=∑ₙ₌₀ ᪲a_nxⁿ that is convergent for real x in the open interval (0, 1), the value lim _(x→1⁻)∑ₙ₌₀ ᪲a_nxⁿ, which is assigned to f(1)=∑ₙ₌₀ ᪲a_n by the Abel summation method (or A-method)."
      ],
      "links": [
        [
          "mathematical analysis",
          "mathematical analysis"
        ],
        [
          "power series",
          "power series"
        ],
        [
          "convergent",
          "convergent"
        ],
        [
          "real",
          "real"
        ],
        [
          "open interval",
          "open interval"
        ],
        [
          "Abel summation method",
          "Abel summation method"
        ],
        [
          "A-method",
          "A-method"
        ]
      ],
      "raw_glosses": [
        "(mathematical analysis) Given a power series f(x)=∑ₙ₌₀ ᪲a_nxⁿ that is convergent for real x in the open interval (0, 1), the value lim _(x→1⁻)∑ₙ₌₀ ᪲a_nxⁿ, which is assigned to f(1)=∑ₙ₌₀ ᪲a_n by the Abel summation method (or A-method)."
      ],
      "topics": [
        "mathematical-analysis",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "Niels Henrik Abel"
      ]
    }
  ],
  "word": "Abel sum"
}