# convergence of Riemann zeta series

The series

$\sum _{n=1}^{\mathrm{\infty}}}{\displaystyle \frac{1}{{n}^{s}}$ | (1) |

converges absolutely for all $s$ with real part^{} greater than 1.

Proof. Let $s=\sigma +it$ where $\sigma $ and $t$ are real numbers and $\sigma >1$. Then

$$\left|\frac{1}{{n}^{s}}\right|=\frac{1}{|{e}^{s\mathrm{log}n}|}=\frac{1}{{e}^{\sigma \mathrm{log}n}}=\frac{1}{{n}^{\sigma}}.$$ |

Since the series ${\sum}_{n=1}^{\mathrm{\infty}}\frac{1}{{n}^{\sigma}}$ converges, by the $p$-test (http://planetmath.org/PTest), for $\sigma >1$, we conclude that the series (1) is absolutely convergent in the half-plane $\sigma >1$.

Title | convergence of Riemann zeta series |
---|---|

Canonical name | ConvergenceOfRiemannZetaSeries |

Date of creation | 2015-08-22 13:15:14 |

Last modified on | 2015-08-22 13:15:14 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 8 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 11M06 |

Classification | msc 30A99 |

Related topic | ModulusOfComplexNumber |

Related topic | ComplexExponentialFunction |