English Dictionary

binomial series

noun

(mathematics) 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!).

The binomial series is a mathematical series that expands a binomial expression.

意味（1）

(mathematics) 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.

意味（2）

(mathematics, loosely) 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.