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

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day! A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

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

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.

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

dCode retains ownership of the source code of the script Harmonic Number online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be given for free. To download the online Harmonic Number script for offline use on PC, iPhone or Android, ask for price quote on contact page !