# Taylor’s theorem

## 1 Taylor’s Theorem

Let $f$ be a function which is defined on the interval $(a,b)$ and suppose the $n$th derivative ${f}^{(n)}$ exists on $(a,b)$. Then for all $x$ and ${x}_{0}$ in $(a,b)$,

$${R}_{n}(x)=\frac{{f}^{(n)}(y)}{n!}{(x-{x}_{0})}^{n}$$ |

with $y$ strictly between $x$ and ${x}_{0}$ ($y$ depends on the choice of $x$). ${R}_{n}(x)$ is the $n$th remainder of the Taylor series^{} for $f(x)$.

Title | Taylor’s theorem |
---|---|

Canonical name | TaylorsTheorem |

Date of creation | 2013-03-22 11:56:53 |

Last modified on | 2013-03-22 11:56:53 |

Owner | Andrea Ambrosio (7332) |

Last modified by | Andrea Ambrosio (7332) |

Numerical id | 11 |

Author | Andrea Ambrosio (7332) |

Entry type | Theorem |

Classification | msc 41A58 |

Related topic | TaylorSeries |