Definition:Harmonic Numbers

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This page is about Harmonic Numbers. For other uses, see Harmonic.


The harmonic numbers are denoted $H_n$ and are defined for positive integers $n$:

$\ds \forall n \in \Z, n \ge 0: H_n = \sum_{k \mathop = 1}^n \frac 1 k$

From the definition of vacuous summation it is clear that $H_0 = 0$.

General Harmonic Numbers

Let $r \in \R_{>0}$.

For $n \in \N_{> 0}$ the harmonic numbers order $r$ are defined as follows:

$\ds \map {H^{\paren r} } n = \sum_{k \mathop = 1}^n \frac 1 {k^r}$


There appears to be no standard notation for the harmonic numbers.

The notation given here, and used on $\mathsf{Pr} \infty \mathsf{fWiki}$ throughout, is an adaptation for $\mathsf{Pr} \infty \mathsf{fWiki}$ of an idea by Donald E. Knuth, where he used $H_n^{\paren r}$.

Knuth's notation proves unwieldy when extended to the complex numbers, and so we have adopted the more conventional mapping notation $\harm r n$ and hence $\harm r z$.

Ramanujan used $\ds \map {\phi_r} n$.

Other notations that can also be found in the literature include $h_n$, $S_n$ and $\map \psi {n + 1} + \gamma$.


Harmonic Number $H_0$

$H_0 = 0$

Harmonic Number $H_1$

$H_1 = 1$

Harmonic Number $H_2$

$H_2 = \dfrac 3 2$

Harmonic Number $H_3$

$H_3 = \dfrac {11} 6$

Harmonic Number $H_4$

$H_4 = \dfrac {25} {12}$

Harmonic Number $H_5$

$H_5 = \dfrac {137} {60}$

Harmonic Number $H_{10000}$

To $15$ decimal places:

$H_{10000} \approx 9 \cdotp 78760 \, 60360 \, 44382 \, \ldots$

Also see

  • Results about harmonic numbers can be found here.