# Gregory series

The Gregory series is an alternating sum whose value is a quarter that of $\pi $:

$$\frac{\pi}{4}=\sum _{i=0}^{\mathrm{\infty}}{(-1)}^{i}\frac{1}{2i+1}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\mathrm{\dots}$$ |

(The approximate decimal value of this expression is 0.7853981633974483…)

More generally, a Gregory series for a given $n$ is

$$\sum _{i=0}^{\mathrm{\infty}}{(-1)}^{i}\frac{{n}^{2i+1}}{2i+1}.$$ |

The Gregory series is named after the Scottish astronomer and astrologer James Gregory.

Title | Gregory series |

Canonical name | GregorySeries |

Date of creation | 2013-03-22 17:35:33 |

Last modified on | 2013-03-22 17:35:33 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 4 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11-00 |

Classification | msc 51-00 |

Classification | msc 01A16 |

Classification | msc 01A20 |

Classification | msc 01A25 |

Classification | msc 01A32 |

Classification | msc 01A40 |

Related topic | TaylorSeriesOfArcusTangent |