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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) : Mathematics

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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:

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

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

The following recurrence formula can also be applied:

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

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

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

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

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

The first harmonic numbers are:

 1 1/1 1 2 3/2 1.5 3 11/6 1.83333 4 25/12 2.08333 5 137/60 2.28333 6 49/20 2,45 7 363/140 2,59286 8 761/280 2,71786 9 7129/2520 2,82896 10 2,92897 100 5,18738 1000 7,48547 10000 9,78761 100000 12,09015 1000000 14,39272 10000000 16,69531 100000000 18,99790 1000000000 21,30048