# example of ratio test

Consider the sequence given by ${a}_{n}={x}^{n}$ (geometric progression) where $$. Then the series

$$\sum _{j=0}^{\mathrm{\infty}}{a}_{n}$$ |

converges. To see this, we can use the ratio test^{}. We need to consider the sequence $|{a}_{n+1}/{a}_{n}|$. But for any $n\ge 0$ we have (when $x\ne 0$)

$$ |

and therefore the series converges. The ratio test and the previous argument shows that the geometric series^{} diverges for $|x|>1$.

Title | example of ratio test |
---|---|

Canonical name | ExampleOfRatioTest |

Date of creation | 2013-03-22 15:03:20 |

Last modified on | 2013-03-22 15:03:20 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 9 |

Author | drini (3) |

Entry type | Example |

Classification | msc 26A06 |

Classification | msc 40A05 |