About: Binomial series

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In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like for a nonnegative integer . Specifically, the binomial series is the Taylor series for the function centered at , where and . Explicitly, where the power series on the right-hand side of is expressed in terms of the (generalized) binomial coefficients

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rdfs:label	Binomial series (en) متسلسلة ذات حدين (ar) Binomische Reihe (de) Serie binomial (es) Formule du binôme généralisée (fr) 이항 급수 (ko) 二項級数 (ja) Série binomial (pt) Биномиальный ряд (ru)
rdfs:comment	في الرياضيات، متسلسلة ذات حدين هي متسلسلة تايلور في النقطة x = 0 للدالة f(x) = (1 + x) α حيث α ∈ C هو عدد عقدي ما. (ar) In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like for a nonnegative integer . Specifically, the binomial series is the Taylor series for the function centered at , where and . Explicitly, where the power series on the right-hand side of is expressed in terms of the (generalized) binomial coefficients (en) La serie binomial es la serie de Taylor para una función dada por , donde es un número complejo arbitrario. Explícitamente, y la serie binomial es la serie de potencias en el lado derecho de (1), expresada en términos de coeficientes binomiales (generalizados) (es) La formule du binôme généralisée permet de développer une puissance complexe d'une somme de deux termes sous forme d'une somme de série et généralise la formule du binôme de Newton et celle du binôme négatif. Dans le cas d'un exposant rationnel, elle a été énoncée sans démonstration par Newton dans ses Principia Mathematica en 1687, puis prouvée par Euler en 1773. (fr) 数学の特に初等解析学における二項級数（にこうきゅうすう、英: binomial series）は二項式の冪（べき）のマクローリン級数を言う。 (ja) 해석학에서 이항 급수(二項級數, 영어: binomial series)는 이항 계수를 계수로 하는 멱급수이다. 이항식의 거듭제곱의 매클로린 급수이다. 이항 정리의 일반화이다. (ko) Биномиальный ряд — это Ряд Тейлора для функции , заданной выражением где является произвольным комплексным числом, а \|x\| < 1. Ряд в явном виде, и биномиальный ряд справа в формуле является степенным рядом, выраженном в терминах (обобщённых) биномиальных коэффициентов (ru) Em matemática, série binomial é uma série de potências do tipo: se e para todo número real . Exemplo: Achar uma representação de , em séries de potências. Solução: Para , obtemos: que pode ser escrito como: +... (pt) Die binomische Reihe ist eine Potenzreihe, die sich bei einer Verallgemeinerung des binomischen Lehrsatzes auf Potenzen mit reellen oder komplexen Exponenten ergibt: Ist der Exponent eine natürliche Zahl, so bricht die Reihe nach dem Glied mit ab und ist daher dann nur eine endliche Summe. Die Koeffizienten der binomischen Reihe sind die Binomialkoeffizienten, deren Name vom Auftreten im binomischen Lehrsatz abgeleitet ist. Für sie gilt mit der fallenden Faktorielle , wobei für das leere Produkt den Wert 1 zugewiesen bekommt. (de)
dcterms:subject	Factorial and binomial topics Mathematical series Complex analysis Real analysis
Wikipage page ID	696619 (xsd:integer)
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Link from a Wikipage to another Wikipage	Power series Binomial coefficient Binomial theorem Derivative John Wallis Factorial and binomial topics Complex number Continuous function Holomorphic function Mathematics Function (mathematics) Gamma function Conditional convergence Convergent series Crelle's Journal Harmonic series (mathematics) Divergent series Niels Henrik Abel Mathematical series Isaac Newton Taylor series Table of Newtonian series Area Asymptotic analysis Complex analysis Abel's theorem Absolute convergence Real analysis Landau notation Binomial approximation If and only if Integer Ordinary differential equation Radius of convergence Ratio test Multiset coefficient Binomial formula
Link from a Wikipage to an external page	http://www.bibnum.education.fr/math%C3%A9matiques/alg%C3%A9bre/niels-abel-et-les-crit%C3%A8res-de-convergence https://www.quantamagazine.org/how-isaac-newton-discovered-the-binomial-power-series-20220831/%7Ctitle=How
sameAs	Binomial series Binomial series Binomial series Binomial series Binomial series Binomial series Binomial series Binomial series Binomial series Binomial series Binomial series Binomial series Binomial series Binomial series Binomial series Binomial series Binomial series Binomial series
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