binomial formula BinomialFormula 2002-02-19 12:43:45 2006-03-19 09:36:07 Theorem \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand{\reals}{\mathbb{R}} \newcommand{\natnums}{\mathbb{N}} \newcommand{\cnums}{\mathbb{C}} \newcommand{\lp}{\left(} \newcommand{\rp}{\right)} \newcommand{\lb}{\left[} \newcommand{\rb}{\right]} \newcommand{\supth}{^{\text{th}}} \newtheorem{proposition}{Proposition} The binomial formula gives the power series expansion of the $p\supth$ power function. The power $p$ can be an integer, rational, real, or even a complex number. The formula is \begin{align*} (1+x)^p &= \sum_{n=0}^\infty \frac{p^{\underline{n}}}{n!} \, x^n\\ &= \sum_{n=0}^\infty \binom{p}{n} x^n \end{align*} where $p^{\underline{n}}= p(p-1)\ldots (p-n+1)$ denotes the falling factorial, and where $\binom{p}{n}$ denotes the generalized binomial coefficient. For $p=0,1,2,\ldots$ the power series reduces to a polynomial, and we obtain the usual binomial theorem. For other values of $p$, the radius of convergence of the series is $1$; the right-hand series converges pointwise for all complex $|x|<1$ to the value on the left side. Also note that the binomial formula is valid at $x=\pm 1$, but for certain values of $p$ only. Of course, we have convergence if $p$ is a natural number. Furthermore, for $x=1$ and real $p$, we have absolute convergence if $p>0$, and conditional convergence if $-1<p<0$. For $x=-1$ we have absolute convergence for $p>0$.