Binomial series

English

Noun

1. Template:Lb The Maclaurin series expansion of the function f(x) = (1 + x)^α, for arbitrary complex α; the series ${\sum }_{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{\alpha }{k})x^k$, where $(\genfrac{}{}{0ex}{}{\alpha }{k})=\frac{\alpha (\alpha -1)(\alpha -2)\dots (\alpha -k+1)}{k!}$.
   • 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.

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2. Template:Lb The binomial theorem.
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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 $(\genfrac{}{}{0ex}{}{\alpha }{k})$, as defined above, is applicable for the coefficients, which are called Template:M. (Note that the binomial theorem treats the slightly different form (x + y)ⁿ, which does not directly generalise to an infinite series.)

Hypernyms

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Translations

• Finnish: Template:T
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Anagrams

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