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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

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Nth Harmonic Number Calculator

H(N) = 1+1/2+1/3+...+1/N


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

Answers to Questions

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 logarithmhref 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:

11/1
12
3/21.5
311/6
1.833334
25/122.08333
5137/60
2.283336
49/202,45
7363/140
2,592868
761/2802,71786
97129/2520
2,8289610
2,92897
100
5,187381000
7,48547
10000
9,78761100000
12,09015
1000000
14,3927210000000
16,69531
100000000
18,997901000000000
21,30048

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