# a series related to harmonic series

The series

$\sum _{n=1}^{\mathrm{\infty}}}{\displaystyle \frac{1}{n\sqrt[n]{n}}}={\displaystyle \sum _{n=1}^{\mathrm{\infty}}}{\displaystyle \frac{1}{{n}^{1+\frac{1}{n}}}$ | (1) |

is divergent. In fact, since for every positive integer n, one has ${2}^{n}>n$, i.e. $$, any of the series satisfies

$$\frac{1}{n\sqrt[n]{n}}>\frac{1}{2n}.$$ |

Because the harmonic series and therefore also ${\sum}_{1}^{\mathrm{\infty}}\frac{1}{2n}$ diverges, the comparison test^{} implies that the series (1) diverges.

Title | a series related to harmonic series |
---|---|

Canonical name | ASeriesRelatedToHarmonicSeries |

Date of creation | 2013-03-22 17:56:40 |

Last modified on | 2013-03-22 17:56:40 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 5 |

Author | pahio (2872) |

Entry type | Example |

Classification | msc 40A05 |

Related topic | PTest |

Related topic | RaabesCriteria |