# binomial formula

The binomial formula gives the power series expansion of the $p^{\text{th}}$ power function. The power $p$ can be an integer, rational, real, or even a complex number. The formula is

 $\displaystyle(1+x)^{p}$ $\displaystyle=\sum_{n=0}^{\infty}\frac{p^{\underline{n}}}{n!}\,x^{n}$ $\displaystyle=\sum_{n=0}^{\infty}\binom{p}{n}x^{n}$

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. For $x=-1$ we have absolute convergence for $p>0$.

Title binomial formula BinomialFormula 2013-03-22 12:23:52 2013-03-22 12:23:52 rmilson (146) rmilson (146) 11 rmilson (146) Theorem msc 26A06 Newton’s binomial series BinomialTheorem GeneralizedBinomialCoefficients