# binomial formula

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

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