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Concept
Binomial series
Summary
In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like (1+x)^n for a nonnegative integer n. Specifically, the binomial series is the Taylor series for the function f(x)=(1+x)^{\alpha} centered at x = 0, where \alpha \in \Complex and |x| < 1. Explicitly,
where the power series on the right-hand side of () is expressed in terms of the (generalized) binomial coefficients
:\binom{\alpha}{k} := \frac{\alpha (\alpha-1) (\alpha-2) \cdots (\alpha-k+1)}{k!}.
Special cases
If α is a nonnegative integer n, then the (n + 2)th term and all later terms in the series are 0, since each contains a factor (n − n); thus in this case the series is finite and gives the algebraic binomial formula.
Closely related is the negative binomial series defined by the Taylor series for the function g(x)=(
Official source
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