Binomial series
Taylor 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 {\displayst...
Nº Q833480 ★
Common · Knowledge
Binomial series
Taylor 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 {\displayst...
From Wikipedia
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 }} . It converges when | x | < 1 {\displaystyle |x|<1} . If α is a nonnegative integer n then the xn + 1 term and all later terms in the series are 0, since each contains a factor of (n − n). In this case, the series is a finite polynomial, equivalent to the binomial formula.
Text: Wikipédia, CC BY-SA 4.0. ·
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