binomial series

binomial series

English

Noun

binomial series (plural binomial series)

1. (mathematics) The Maclaurin series expansion of the function f(x) = (1 + x)^α, for arbitrary complex α; the series ${\\displaystyle \\textstyle \\sum _{k=0}^{\\infty }{\\alpha \\choose k}x^{k}}$, where ${\\displaystyle \\textstyle {\\alpha \\choose k}={\\frac {\\alpha (\\alpha -1)(\\alpha -2)\\dots (\\alpha -k+1)}{k!}}}$.
   • 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.

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

     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.

2. (mathematics, loosely) The binomial theorem.
   • 2000, W. Bolton, Mathematics for Engineering, page 92,

     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.

   • 2017, John Bird, Higher Engineering Mathematics, page 60,

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

Usage notes

The infinite series is a direct generalisation of the (finite) binomial theorem expansion of (1 + x)ⁿ (n a positive integer): in both cases, the notation ${\\displaystyle \\textstyle {\\alpha \\choose k}}$, as defined above, is applicable for the coefficients, which are called binomial coefficients. (Note that the binomial theorem treats the slightly different form (x + y)ⁿ, which does not directly generalise to an infinite series.)

Hypernyms

• (Maclaurin series expansion of (1 + x)^α): Maclaurin series, power series Taylor series

Translations

Maclaurin series expansion of (1 + x)^α