# 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

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

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

where ${p}^{\underset{\xaf}{n}}=p(p-1)\mathrm{\dots}(p-n+1)$ denotes the falling
factorial^{}, and where $\left(\genfrac{}{}{0pt}{}{p}{n}\right)$ denotes the generalized binomial
coefficient.

For $p=0,1,2,\mathrm{\dots}$ 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 $$ to the value on the left
side. Also note that the binomial formula is valid at $x=\pm 1$, but
for certain values of $p$ only. Of course, we have convergence if $p$
is a natural number^{}. Furthermore, for $x=1$ and real $p$, we have
absolute convergence^{} if $p>0$, and conditional convergence if
$$. 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 |