# radius of convergence

To the power series^{}

$$\sum _{k=0}^{\mathrm{\infty}}{a}_{k}{(x-{x}_{0})}^{k}$$ | (1) |

there exists a number $r\in [0,\mathrm{\infty}]$, its *radius of convergence ^{}*, such that the series converges absolutely for all (real or complex) numbers $x$ with $$ and diverges whenever $|x-{x}_{0}|>r$. This is known as Abel’s theorem on power series. (For $|x-{x}_{0}|=r$ no general statements can be made.)

The radius of convergence is given by:

$$r=\underset{k\to \mathrm{\infty}}{lim\; inf}\frac{1}{\sqrt[k]{|{a}_{k}|}}$$ | (2) |

and can also be computed as

$$r=\underset{k\to \mathrm{\infty}}{lim}\left|\frac{{a}_{k}}{{a}_{k+1}}\right|,$$ | (3) |

if this limit exists.

It follows from the Weierstrass $M$-test (http://planetmath.org/WeierstrassMTest) that for any radius ${r}^{\prime}$ smaller than the radius of convergence, the power series converges uniformly within the closed disk of radius ${r}^{\prime}$.

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

Canonical name | RadiusOfConvergence |

Date of creation | 2013-03-22 12:32:59 |

Last modified on | 2013-03-22 12:32:59 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 13 |

Author | PrimeFan (13766) |

Entry type | Theorem |

Classification | msc 40A30 |

Classification | msc 30B10 |

Synonym | Abel’s theorem on power series |

Related topic | ExampleOfAnalyticContinuation |

Related topic | NielsHenrikAbel |