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

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 $p>0$ .