Binomial Series

The binomial series is the extension of the binomial formula to exponents that are not whole numbers. For a complex number alpha, it expands the quantity one plus x, raised to the power alpha, as an infinite sum built from generalized binomial coefficients formed from alpha rather than a whole number: one plus alpha times x plus alpha times alpha minus one over two factorial times x squared, and so on. This series is the Maclaurin expansion of the function one plus x to the alpha, and it converges whenever the absolute value of x is less than one. When alpha is a non-negative whole number the series becomes the ordinary finite binomial theorem, because every term past a certain point becomes zero. Because Isaac Newton worked on this generalization early on, it is sometimes called Newton binomial theorem.