# proof of radius of convergence

According to Cauchy’s root test^{} a power series^{} is absolutely convergent if

$$ |

This is obviously true if

$$ |

In the same way we see that the series is divergent if

$$|x-{x}_{0}|>\underset{k\to \mathrm{\infty}}{lim\; inf}\frac{1}{\sqrt[k]{|{a}_{k}|}},$$ |

which means that the right hand side is the radius of convergence^{} of the power series.
Now from the ratio test^{} we see that the power series is absolutely convergent if

$$ |

Again this is true if

$$ |

The series is divergent if

$$|x-{x}_{0}|>\underset{k\to \mathrm{\infty}}{lim}\left|\frac{{a}_{k}}{{a}_{k+1}}\right|,$$ |

as follows from the ratio test in the same way. So we see that in this way too we can the radius of convergence.

Title | proof of radius of convergence |
---|---|

Canonical name | ProofOfRadiusOfConvergence |

Date of creation | 2013-03-22 13:21:50 |

Last modified on | 2013-03-22 13:21:50 |

Owner | mathwizard (128) |

Last modified by | mathwizard (128) |

Numerical id | 6 |

Author | mathwizard (128) |

Entry type | Proof |

Classification | msc 40A30 |

Classification | msc 30B10 |