binomial series

noun

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!).

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Meanings

nounbinomial series (plural binomial series)
  1. mathematicsThe 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!).

    2010, James Stewart, Calculus: Concepts and Contexts, Cengage Learning, page 612, Thus, by the Ratio Test, the binomial series converges if |x| < 1 and diverges if |x| > 1.

    1 more example

    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.

    2015, Hans-Heinrich Körle, Infinite Series in a History of Analysis: Stages up to the Verge of Summability, Walter de Gruyter, page 36
  2. broadlymathematicsThe binomial theorem.

    The binomial series or binomial theorem is a formula for raising a binomial expression to any power without lengthy multiplication.

    2017, John Bird, Higher Engineering Mathematics, page 60
    1 more example

    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.

    2000, W. Bolton, Mathematics for Engineering, page 92

Translations

Finnish binomisarja
  • binomisarjaMaclaurin series expansion of (1 + x)α
German binomische Reihe
  • binomische ReiheMaclaurin series expansion of (1 + x)α
Cite this entry
APA

binomial series. (n.d.). In WatkinsX Dictionary. Retrieved November 3, 2026, from https://dictionary.watkinsx.com/word/binomial%20series

MLA

“binomial series.” WatkinsX Dictionary, https://dictionary.watkinsx.com/word/binomial%20series. Accessed November 3, 2026.

Chicago

WatkinsX Dictionary, s.v. “binomial series,” accessed November 3, 2026, https://dictionary.watkinsx.com/word/binomial%20series.

Source: binomial series on Wiktionary, CC BY-SA 4.0.