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

${(1+x)}^p$$={\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{p^{\underset{¯}{n}}}{n!}}{x}^n$

$={\displaystyle \sum _{n=0}^{\mathrm{\infty }}}\left({\displaystyle \genfrac{}{}{0pt}{}{p}{n}}\right)x^n$

where $p^{\underset{¯}{n}}=p(p-1)\mathrm{\dots }(p-n+1)$ denotes the fallingfactorial, and where $\left(\genfrac{}{}{0pt}{}{p}{n}\right)$ denotes the generalized binomialcoefficient.

For $p=0,1,2,\mathrm{\dots }$ the power series reduces to a polynomial, and weobtain the usual binomial theorem. For other values of $p$, theradius of convergence of the series is $1$; the right-hand seriesconverges pointwise for all complex to the value on the leftside. Also note that the binomial formula is valid at $x=\pm 1$, butfor certain values of $p$ only. Of course, we have convergence if $p$is a natural number. Furthermore, for $x=1$ and real $p$, we haveabsolute convergence if $p>0$, and conditional convergence if. For $x=-1$ we have absolute convergence for $p>0$.