Meaning of binomial series | Babel Free

Definitions

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!).
2. The binomial theorem.
   broadly

Equivalents

Examples

“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.”

“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.”

“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.”

CEFR level