## You are here

Homeharmonic series

## Primary tabs

# harmonic series

The *harmonic series* is

$h=\sum_{{n=1}}^{\infty}\frac{1}{n}$ |

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

$\int_{{1}}^{\infty}\frac{1}{x}\;dx.$ |

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

$h_{p}=\sum_{{n=1}}^{\infty}\frac{1}{n^{p}}$ |

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 $p<1$ 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 $\infty$, 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 $p$.

## Mathematics Subject Classification

40A05*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections