"binomial series" meaning in All languages combined

{
  "forms": [
    {
      "form": "binomial series",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {
        "1": "binomial series"
      },
      "expansion": "binomial series (plural binomial series)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Mathematics",
          "orig": "en:Mathematics",
          "parents": [
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "text": "2010, James Stewart, Calculus: Concepts and Contexts, Cengage Learning, page 612,\nThus, by the Ratio Test, the binomial series converges if |x| < 1 and diverges if |x| > 1."
        },
        {
          "ref": "2015, Hans-Heinrich Körle, Infinite Series in a History of Analysis: Stages up to the Verge of Summability, Walter de Gruyter, page 36",
          "text": "While applying his willful kind of calculus to Newton's binomial series, Euler not only constructed the exponential function, but found out that its inverse had already been in the minds of the 17th century and thus detected the very nature of logarithms.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "The Maclaurin series expansion of the function f(x) = (1 + x)^α, for arbitrary complex α; the series ∑ₖ₌₀ ᪲α choose kxᵏ, where α choose k=(α(α-1)(α-2)…(α-k+1))/(k!)."
      ],
      "hypernyms": [
        {
          "_dis1": "71 29",
          "sense": "Maclaurin series, power series Taylor series",
          "word": "Maclaurin series expansion of (1 + x)α"
        }
      ],
      "id": "en-binomial_series-en-noun-JxyJoouj",
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "Maclaurin series",
          "Maclaurin series"
        ],
        [
          "function",
          "function"
        ],
        [
          "complex",
          "complex"
        ]
      ],
      "raw_glosses": [
        "(mathematics) The Maclaurin series expansion of the function f(x) = (1 + x)^α, for arbitrary complex α; the series ∑ₖ₌₀ ᪲α choose kxᵏ, where α choose k=(α(α-1)(α-2)…(α-k+1))/(k!)."
      ],
      "topics": [
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "_dis1": "76 24",
          "code": "fi",
          "lang": "Finnish",
          "sense": "Maclaurin series expansion of (1 + x)α",
          "word": "binomisarja"
        },
        {
          "_dis1": "76 24",
          "code": "de",
          "lang": "German",
          "sense": "Maclaurin series expansion of (1 + x)α",
          "tags": [
            "feminine"
          ],
          "word": "binomische Reihe"
        }
      ]
    },
    {
      "categories": [
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Mathematics",
          "orig": "en:Mathematics",
          "parents": [
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        },
        {
          "_dis": "31 69",
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w+disamb"
        }
      ],
      "examples": [
        {
          "ref": "2000, W. Bolton, Mathematics for Engineering, page 92",
          "text": "A particular useful series is termed the binomial series (or binomial theorem) and is frequently used to simplify engineering expressions. In this chapter we consider the arithmetic, geometric and binomial series.",
          "type": "quotation"
        },
        {
          "ref": "2017, John Bird, Higher Engineering Mathematics, page 60",
          "text": "The binomial series or binomial theorem is a formula for raising a binomial expression to any power without lengthy multiplication.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "The binomial theorem."
      ],
      "id": "en-binomial_series-en-noun-QiLvUNG5",
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "binomial theorem",
          "binomial theorem"
        ]
      ],
      "raw_glosses": [
        "(mathematics, loosely) The binomial theorem."
      ],
      "tags": [
        "broadly"
      ],
      "topics": [
        "mathematics",
        "sciences"
      ]
    }
  ],
  "wikipedia": [
    "binomial series"
  ],
  "word": "binomial series"
}

{
  "categories": [
    "English countable nouns",
    "English entries with incorrect language header",
    "English lemmas",
    "English multiword terms",
    "English nouns"
  ],
  "forms": [
    {
      "form": "binomial series",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {
        "1": "binomial series"
      },
      "expansion": "binomial series (plural binomial series)",
      "name": "en-noun"
    }
  ],
  "hypernyms": [
    {
      "sense": "Maclaurin series, power series Taylor series",
      "word": "Maclaurin series expansion of (1 + x)α"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "English terms with quotations",
        "en:Mathematics"
      ],
      "examples": [
        {
          "text": "2010, James Stewart, Calculus: Concepts and Contexts, Cengage Learning, page 612,\nThus, by the Ratio Test, the binomial series converges if |x| < 1 and diverges if |x| > 1."
        },
        {
          "ref": "2015, Hans-Heinrich Körle, Infinite Series in a History of Analysis: Stages up to the Verge of Summability, Walter de Gruyter, page 36",
          "text": "While applying his willful kind of calculus to Newton's binomial series, Euler not only constructed the exponential function, but found out that its inverse had already been in the minds of the 17th century and thus detected the very nature of logarithms.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "The Maclaurin series expansion of the function f(x) = (1 + x)^α, for arbitrary complex α; the series ∑ₖ₌₀ ᪲α choose kxᵏ, where α choose k=(α(α-1)(α-2)…(α-k+1))/(k!)."
      ],
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "Maclaurin series",
          "Maclaurin series"
        ],
        [
          "function",
          "function"
        ],
        [
          "complex",
          "complex"
        ]
      ],
      "raw_glosses": [
        "(mathematics) The Maclaurin series expansion of the function f(x) = (1 + x)^α, for arbitrary complex α; the series ∑ₖ₌₀ ᪲α choose kxᵏ, where α choose k=(α(α-1)(α-2)…(α-k+1))/(k!)."
      ],
      "topics": [
        "mathematics",
        "sciences"
      ]
    },
    {
      "categories": [
        "English terms with quotations",
        "en:Mathematics"
      ],
      "examples": [
        {
          "ref": "2000, W. Bolton, Mathematics for Engineering, page 92",
          "text": "A particular useful series is termed the binomial series (or binomial theorem) and is frequently used to simplify engineering expressions. In this chapter we consider the arithmetic, geometric and binomial series.",
          "type": "quotation"
        },
        {
          "ref": "2017, John Bird, Higher Engineering Mathematics, page 60",
          "text": "The binomial series or binomial theorem is a formula for raising a binomial expression to any power without lengthy multiplication.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "The binomial theorem."
      ],
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "binomial theorem",
          "binomial theorem"
        ]
      ],
      "raw_glosses": [
        "(mathematics, loosely) The binomial theorem."
      ],
      "tags": [
        "broadly"
      ],
      "topics": [
        "mathematics",
        "sciences"
      ]
    }
  ],
  "translations": [
    {
      "code": "fi",
      "lang": "Finnish",
      "sense": "Maclaurin series expansion of (1 + x)α",
      "word": "binomisarja"
    },
    {
      "code": "de",
      "lang": "German",
      "sense": "Maclaurin series expansion of (1 + x)α",
      "tags": [
        "feminine"
      ],
      "word": "binomische Reihe"
    }
  ],
  "wikipedia": [
    "binomial series"
  ],
  "word": "binomial series"
}