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 $p$ .