Search for a tool
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. 1 + 1/2 + 1/3 + … + 1/n

Results

Harmonic Number -

Tag(s) : Series

Share
dCode and more

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!

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Feedback and suggestions are welcome so that dCode offers the best 'Harmonic Number' tool for free! Thank you!

# Harmonic Number

## Nth Harmonic Number Calculator

$$H(N) = 1+1/2+1/3+…+1/N$$

## Reciprocal Harmonic Value

$$H(N) = x \iff N = ?$$

### What is a harmonic number? (Definition)

Harmonic numbers are real numbers present in the harmonic series $H_n$ (which uses the sum of the inverse of non-zero natural integers).

### How to calculate a harmonic number?

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

Example: $H_2 = 1+\frac{1}{2} = \frac{3}{2} = 1.5$

The following recurrence formula can also be applied to get a series:

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

When $n$ is very big, an approximation based on the natural logarithm can be useful to speed up the calculations:

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

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:

nH(n)≈H(n)
1st harmonic number1/11
2nd harmonic number3/21.5
3rd harmonic number11/61.83333
4th harmonic number25/122.08333
5th harmonic number137/602.28333
6th harmonic number49/202.45
7th harmonic number363/1402.59286
8th harmonic number761/2802.71786
9th harmonic number7129/25202.82896
10th harmonic number2.92897
100th harmonic number5.18738
1000th harmonic number7.48547
10000th harmonic number9.78761
100000th harmonic number12.09015
1000000th harmonic number14.39272
10000000th harmonic number16.69531
100000000th harmonic number18.99790
1000000000th harmonic number21.30048

### Is the Harmonic Series convergent?

No, the harmonic series is an example of a divergent series, the sum of the terms of the series has no finite limit and tends towards infinity.

### What is the relationship between harmonic numbers and the Riemann Zeta function?

The Harmonic series with $n \to \infty$ is a special case of the Riemann Zeta function ζ(s), when $s = 1$.

### How to implement the Harmonic series?

The algorithm for calculating harmonic numbers can use a summation loop // Pseudo-codefunction harmonicNumber(N) { harmonic = 0 for (i = 1; i <= N; i++) { harmonic = harmonic + 1 / i } return harmonic}

## Source code

dCode retains ownership of the "Harmonic Number" source code. Except explicit open source licence (indicated Creative Commons / free), the "Harmonic Number" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Harmonic Number" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Harmonic Number" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.

## Cite dCode

The copy-paste of the page "Harmonic Number" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Harmonic Number on dCode.fr [online website], retrieved on 2024-09-09, https://www.dcode.fr/harmonic-number

## Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!