# proof of binomial formula

Let $p\in \mathbb{R}$ and $$ be given. We wish to show that

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

where ${p}^{\underset{\xaf}{n}}$ denotes the ${n}^{\text{th}}$ falling factorial of $p$.

The convergence of the series in the right-hand side of the above
equation is a straight-forward consequence of the ratio test^{}. Set

$$f(x)={(1+x)}^{p}.$$ |

and note that

$${f}^{(n)}(x)={p}^{\underset{\xaf}{n}}{(1+x)}^{p-n}.$$ |

The desired equality now follows from Taylor’s Theorem. Q.E.D.

Title | proof of binomial formula |
---|---|

Canonical name | ProofOfBinomialFormula |

Date of creation | 2013-03-22 12:24:00 |

Last modified on | 2013-03-22 12:24:00 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 6 |

Author | rmilson (146) |

Entry type | Proof |

Classification | msc 26A06 |