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

$${\int }_1^{\mathrm{\infty }}\frac{1}{x}𝑑x.$$

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 $p$.