binomial series

Noun

binomial series (plural binomial series)

1. 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!).
   • Thus, by the Ratio Test, the binomial series converges if |x| < 1 and diverges if |x| > 1.2010, James Stewart, Calculus: Concepts and Contexts, Cengage Learning, page 612
   • 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. (broadly) The binomial theorem.
   • 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
   • 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