harmonic series

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

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 $p$.

Title harmonic series HarmonicSeries 2013-03-22 13:02:46 2013-03-22 13:02:46 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 40A05 HarmonicNumber PrimeHarmonicSeries SumOfPowers p-series harmonic series of order