harmonic series


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


The harmonic series is a special case of the p-series, hp, which has the form


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 testMathworldPlanetmath, one can often compare a given series with positive terms to some hp.

Remark 1. One could call hp with  p>1  an overharmonic series and hp 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, hp=ζ(p), the Riemann zeta functionDlmfDlmfMathworldPlanetmath.

A famous p-series is h2 (or ζ(2)), which converges to π26. In general no p-series of odd p has been solved analytically.

A p-series which is not summed to , but instead is of the form


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