binomial series

Noun

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

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 $$\textstyle{\alpha \choose k}$$, as defined above, is applicable for the coefficients, which are called. (Note that the binomial theorem treats the slightly different form (x + y)n, which does not directly generalise to an infinite series.)

Translations

 * Finnish: binomisarja
 * German: binomische Reihe