# harmonic series

The *harmonic series ^{}* is

$$h=\sum _{n=1}^{\mathrm{\infty}}\frac{1}{n}$$ |

The harmonic series is known to diverge. This can be proven via the integral test^{}; compare $h$ with

$${\int}_{1}^{\mathrm{\infty}}\frac{1}{x}\mathit{d}x.$$ |

The harmonic series is a special case of the *$p$-series*, ${h}_{p}$, which has the form

$${h}_{p}=\sum _{n=1}^{\mathrm{\infty}}\frac{1}{{n}^{p}}$$ |

where $p$ is some positive real number. The series is known to converge (leading to the p-series test for series convergence) iff $p>1$. In using the comparison test^{}, one can often compare a given series with positive terms to some ${h}_{p}$.

Remark 1. One could call ${h}_{p}$ with $p>1$ an overharmonic series and ${h}_{p}$ with $$ an underharmonic series; the corresponding names are known at least in Finland.

Remark 2. A $p$-series is sometimes called *a* harmonic series, so that *the* harmonic series is a harmonic series with $p=1$.

For complex-valued $p$, ${h}_{p}=\zeta (p)$, the Riemann zeta function^{}.

A famous $p$-series is ${h}_{2}$ (or $\zeta (2)$), which converges to $\frac{{\pi}^{2}}{6}$. In general no $p$-series of odd $p$ has been solved analytically.

A $p$-series which is not summed to $\mathrm{\infty}$, but instead is of the form

$${h}_{p}(k)=\sum _{n=1}^{k}\frac{1}{{n}^{p}}$$ |

is called a $p$-series (or a harmonic series) of order $k$ of $p$.

Title | harmonic series |
---|---|

Canonical name | HarmonicSeries |

Date of creation | 2013-03-22 13:02:46 |

Last modified on | 2013-03-22 13:02:46 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 8 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 40A05 |

Related topic | HarmonicNumber |

Related topic | PrimeHarmonicSeries |

Related topic | SumOfPowers |

Defines | p-series |

Defines | harmonic series of order |