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

Harmonic Number - dCode

Tag(s) : Series

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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: $$ H_n = \int_0^1 \frac{1 - x^n}{1 - x}\,dx $$

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 |

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*Harmonic Number* on dCode.fr [online website], retrieved on 2022-08-08,

series,harmonic,number,inverse,euler

https://www.dcode.fr/harmonic-number

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