---
tag: 0D2Y6G4C4S
kind: definition
name: Harmonic number
msc: [11B83]
defines: 0NANDX7V02
sources:
  - url: https://en.wikipedia.org/wiki/Harmonic_number
    locator: lead section
---

## Definition

Let $n$ be a natural number.
The $n$-th harmonic number is the rational number
$$H_n = \sum_{k=1}^{n} \frac{1}{k} = 1 + \frac{1}{2} + \cdots + \frac{1}{n}.$$

## Remarks

The first values are $H_0 = 0$, $H_1 = 1$, $H_2 = 3/2$, $H_3 = 11/6$ and $H_4 = 25/12$.
The harmonic numbers are the partial sums of the [[harmonic series|0CXTMAY5PX]].
They satisfy $H_n - \ln n \to \gamma$, where $\gamma$ is the [[Euler–Mascheroni constant|00C5P9JZBK]].

## Explanations

### L1 human

The harmonic numbers are the running totals of $1 + 1/2 + 1/3 + \cdots$.
They grow without bound, but very slowly. One needs $n = 12367$ terms to pass $10$.