Harmonic number

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. They satisfy $H_n - \ln n \to \gamma$, where $\gamma$ is the Euler–Mascheroni constant.

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 $10$.