binomial series

Definitions

1. 1

   The Maclaurin series expansion of the function f(x) = (1 + x)^α, for arbitrary complex α

   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!).

   • 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.
2. 2

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

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