# boundedness of terms of power series

Theorem. If the set

$$\{{a}_{0},{a}_{1}c,{a}_{2}{c}^{2},\mathrm{\dots}\}$$ |

of the of a power series^{}

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

at the point $z=c$ is bounded (http://planetmath.org/BoundedInterval), then the power series converges^{}, absolutely (http://planetmath.org/AbsoluteConvergence), for any value $z$ which satisfies

$$ |

*Proof.* By the assumption^{}, there exists a positive number $M$ such that

$$ |

Thus one gets for the coefficients of the series the estimation

$$ |

If now $$, one has

$$ |

and since the geometric series^{} $\sum _{n=0}^{\mathrm{\infty}}}{\left|{\displaystyle \frac{z}{c}}\right|}^{n$ is convergent, then also the real series $\sum _{n=0}^{\mathrm{\infty}}}|{a}_{n}{z}^{n}|$ converges.

Title | boundedness of terms of power series |
---|---|

Canonical name | BoundednessOfTermsOfPowerSeries |

Date of creation | 2013-03-22 18:50:44 |

Last modified on | 2013-03-22 18:50:44 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 5 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 40A30 |

Classification | msc 30B10 |