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Harmonic Number

Tool for calculating the values of the harmonic numbers, ie the values of the nth partial sums of the harmonic series as well as their inverse. The harmonic series is the series of inverses of natural non-zero integers. 1 + 1/2 + 1/3 + ... + 1/n

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Harmonic Number -

Tag(s) : Series

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# Harmonic Number

## Reciproqual Harmonic Value

Tool for calculating the values of the harmonic numbers, ie the values of the nth partial sums of the harmonic series as well as their inverse. The harmonic series is the series of inverses of natural non-zero integers. 1 + 1/2 + 1/3 + ... + 1/n

### How to calculate an harmonic number?

Harmonic numbers are described by the formula: (sum of inverses of natural numbers)

$$H_n = \sum_{k=1}^n \frac{1}{k} = 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$$

Example: $H_3 = 1+\frac{1}{2} = \frac{3}{2} = 1.5$

The following recurrence formula can also be applied to get a series:

$$H_n = H_{n-1} + \frac{1}{n}$$

$H_n$ is called the Harmonic series.

When $n$ is very big, the following approximation using logarithm can be applied

$$\lim_{n \to \infty} H_n = \ln n + \gamma$$

with $\gamma \approx 0.577215665$ the Euler–Mascheroni constant.

There is also a formula based on a integrate calculation: $$H_n = \int_0^1 \frac{1 - x^n}{1 - x}\,dx$$

### What are the first values of the Harmonic Series?

The first harmonic numbers are:

nH(n)≈H(n)
11/11
23/21.5
311/61.83333
425/122.08333
5137/602.28333
649/202.45
7363/1402.59286
8761/2802.71786
97129/25202.82896
102.92897
1005.18738
10007.48547
100009.78761
10000012.09015
100000014.39272
1000000016.69531
10000000018.99790
100000000021.30048

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