Binomial series

The binomial series is the Taylor series for the function f {displaystyle f} given by f ( x ) = ( 1 + x ) α {displaystyle f(x)=(1+x)^{alpha }} , where α ∈ C {displaystyle alpha in mathbb {C} } is an arbitrary complex number. Explicitly, The binomial series is the Taylor series for the function f {displaystyle f} given by f ( x ) = ( 1 + x ) α {displaystyle f(x)=(1+x)^{alpha }} , where α ∈ C {displaystyle alpha in mathbb {C} } is an arbitrary complex number. Explicitly, and the binomial series is the power series on the right hand side of (1), expressed in terms of the (generalized) binomial coefficients 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. The following variant holds for arbitrary complex β, but is especially useful for handling negative integer exponents in (1): To prove it, substitute x = −z in (1) and apply a binomial coefficient identity, which is, Whether (1) converges depends on the values of the complex numbers α and x. More precisely: In particular, if α {displaystyle alpha } is not a non-negative integer, the situation at the boundary of the disk of convergence, | x | = 1 {displaystyle |x|=1} , is summarized as follows: