"formal power series" meaning in English

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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
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          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
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          "langcode": "en",
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      "examples": [
        {
          "text": "1995 Alberto Bertoni, Massimiliano Goldwurm, Giancarlo Mauri, Nicoletta Sabatini, Chapter 5: Counting Techniques for Inclusion, Equivalence, and Membership Problems, Volker Diekert, Grzegorz Rozenberg, The Book of Traces, World Scientific, page 138,\nMoreover, it is useful to observe that in this case the definition of rational formal power series can be simplified: a f.p.s. r∈ℚ⟨!⟨z⟩!⟩ is rational if and only if r is the quotient of two polynomials."
        },
        {
          "text": "1997, Greg Marks, Direct Product and Power Series Formations over 2-Primal Rings, Surender Kumar Jain, S. Tariq Rizvi, Advances in Ring Theory, Springer, page 239,\nWe also show that the ring of formal power series over a 2-primal ring (or even a ring satisfying (PS I)) need not be 2-primal."
        },
        {
          "ref": "2011, Nicholas Loehr, Bijective Combinatorics, Taylor & Francis (CRC Press), page 243",
          "text": "This chapter gives a rigorous development of the algebraic properties of formal power series. Our goal is to extend the familiar operations on polynomial functions (like addition, multiplication, composition, and differentiation) to the setting of formal power series. In certain situations we will even be able to define infinite sums and products of formal power series.[…]In combinatorics, it usually suffices to consider formal power series whose coefficients are integers, rational numbers, or complex numbers.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Any finite or infinite series of the form a_0+a_1x+a_2x²…=∑ᵢa_ixⁱ, where the aᵢ are numbers, but it is understood that no value is assigned to x."
      ],
      "id": "en-formal_power_series-en-noun-hGEyg8I5",
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          "infinite",
          "infinite"
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          "series",
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      ],
      "raw_glosses": [
        "(mathematics, algebra) Any finite or infinite series of the form a_0+a_1x+a_2x²…=∑ᵢa_ixⁱ, where the aᵢ are numbers, but it is understood that no value is assigned to x."
      ],
      "related": [
        {
          "word": "formal Laurent series"
        },
        {
          "word": "Lagrange inversion theorem"
        },
        {
          "word": "Lagrange-Bürmann formula"
        }
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      "wikipedia": [
        "formal power series"
      ]
    }
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  "word": "formal power series"
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{
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      "form": "formal power series",
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "formal Laurent series"
    },
    {
      "word": "Lagrange inversion theorem"
    },
    {
      "word": "Lagrange-Bürmann formula"
    }
  ],
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      "examples": [
        {
          "text": "1995 Alberto Bertoni, Massimiliano Goldwurm, Giancarlo Mauri, Nicoletta Sabatini, Chapter 5: Counting Techniques for Inclusion, Equivalence, and Membership Problems, Volker Diekert, Grzegorz Rozenberg, The Book of Traces, World Scientific, page 138,\nMoreover, it is useful to observe that in this case the definition of rational formal power series can be simplified: a f.p.s. r∈ℚ⟨!⟨z⟩!⟩ is rational if and only if r is the quotient of two polynomials."
        },
        {
          "text": "1997, Greg Marks, Direct Product and Power Series Formations over 2-Primal Rings, Surender Kumar Jain, S. Tariq Rizvi, Advances in Ring Theory, Springer, page 239,\nWe also show that the ring of formal power series over a 2-primal ring (or even a ring satisfying (PS I)) need not be 2-primal."
        },
        {
          "ref": "2011, Nicholas Loehr, Bijective Combinatorics, Taylor & Francis (CRC Press), page 243",
          "text": "This chapter gives a rigorous development of the algebraic properties of formal power series. Our goal is to extend the familiar operations on polynomial functions (like addition, multiplication, composition, and differentiation) to the setting of formal power series. In certain situations we will even be able to define infinite sums and products of formal power series.[…]In combinatorics, it usually suffices to consider formal power series whose coefficients are integers, rational numbers, or complex numbers.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Any finite or infinite series of the form a_0+a_1x+a_2x²…=∑ᵢa_ixⁱ, where the aᵢ are numbers, but it is understood that no value is assigned to x."
      ],
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      "raw_glosses": [
        "(mathematics, algebra) Any finite or infinite series of the form a_0+a_1x+a_2x²…=∑ᵢa_ixⁱ, where the aᵢ are numbers, but it is understood that no value is assigned to x."
      ],
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      "wikipedia": [
        "formal power series"
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}