binomial formula

The binomial formulaMathworldPlanetmath gives the power seriesMathworldPlanetmath expansion of the pth power function. The power p can be an integer, rational, real, or even a complex numberPlanetmathPlanetmath. The formulaMathworldPlanetmathPlanetmath is

(1+x)p =n=0pn¯n!xn

where pn¯=p(p-1)(p-n+1) denotes the falling factorialMathworldPlanetmath, and where (pn) denotes the generalized binomial coefficient.

For p=0,1,2, the power series reduces to a polynomialPlanetmathPlanetmath, and we obtain the usual binomial theoremMathworldPlanetmath. For other values of p, the radius of convergenceMathworldPlanetmath of the series is 1; the right-hand series convergesPlanetmathPlanetmath pointwisePlanetmathPlanetmath for all complex |x|<1 to the value on the left side. Also note that the binomial formula is valid at x=±1, but for certain values of p only. Of course, we have convergence if p is a natural numberMathworldPlanetmath. Furthermore, for x=1 and real p, we have absolute convergenceMathworldPlanetmath if p>0, and conditional convergence if -1<p<0. For x=-1 we have absolute convergence for p>0.

Title binomial formula
Canonical name BinomialFormula
Date of creation 2013-03-22 12:23:52
Last modified on 2013-03-22 12:23:52
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 11
Author rmilson (146)
Entry type Theorem
Classification msc 26A06
Synonym Newton’s binomial series
Related topic BinomialTheorem
Related topic GeneralizedBinomialCoefficients