Binomial series

Mathematical series

In mathematics, the binomial series is a generalization of the binomial formula to cases where the exponent is not a positive integer: where α {\displaystyle \alpha } is any complex number, and the power series on the right-hand side is expressed in terms of the (generalized) binomial coefficients ( α k) = α ( α − 1) ( α − 2) ⋯ ( α − k + 1) k ! . {\displaystyle {\binom {\alpha }{k}}={\frac {\alpha (\alpha -1)(\alpha -2)\cdots (\alpha -k+1)}{k!}}.} The binomial series is the MacLaurin series for the function f ( x) = ( 1 + x) α {\displaystyle f(x)=(1+x)^{\alpha }} .

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