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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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:

n

H(n)

≈H(n)

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

Source code

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Questions / Comments

Thanks to your feedback and relevant comments, dCode has developed the best 'Harmonic Number' tool, so feel free to write! Thank you!

Thanks to your feedback and relevant comments, dCode has developed the best 'Harmonic Number' tool, so feel free to write! Thank you!