"Abel sum" meaning in All languages combined

{
  "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"
        },
        {
          "kind": "other",
          "name": "English entries with language name categories using raw markup",
          "parents": [
            "Entries with language name categories using raw markup",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "English terms with non-redundant non-automated sortkeys",
          "parents": [
            "Terms with non-redundant non-automated sortkeys",
            "Entry maintenance"
          ],
          "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": "quotation"
        },
        {
          "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": "quotation"
        }
      ],
      "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 entries with language name categories using raw markup",
        "English eponyms",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with non-redundant non-automated sortkeys",
        "English terms with quotations",
        "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": "quotation"
        },
        {
          "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": "quotation"
        }
      ],
      "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"
}