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
Harmonic Number - dCode
Tag(s) : Series
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Harmonic Number
Nth Harmonic Number Calculator
Reciprocal Harmonic Value
Answers to Questions (FAQ)
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:
| n | H(n) | ≈H(n) |
|---|---|---|
| 1st harmonic number | 1/1 | 1 |
| 2nd harmonic number | 3/2 | 1.5 |
| 3rd harmonic number | 11/6 | 1.83333 |
| 4th harmonic number | 25/12 | 2.08333 |
| 5th harmonic number | 137/60 | 2.28333 |
| 6th harmonic number | 49/20 | 2.45 |
| 7th harmonic number | 363/140 | 2.59286 |
| 8th harmonic number | 761/280 | 2.71786 |
| 9th harmonic number | 7129/2520 | 2.82896 |
| 10th harmonic number | 2.92897 | |
| 100th harmonic number | 5.18738 | |
| 1000th harmonic number | 7.48547 | |
| 10000th harmonic number | 9.78761 | |
| 100000th harmonic number | 12.09015 | |
| 1000000th harmonic number | 14.39272 | |
| 10000000th harmonic number | 16.69531 | |
| 100000000th harmonic number | 18.99790 | |
| 1000000000th harmonic number | 21.30048 |
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-code
function harmonicNumber(N) {
harmonic = 0
for (i = 1; i <= N; i++) {
harmonic = harmonic + 1 / i
}
return harmonic
}
Source code
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- Nth Harmonic Number Calculator
- Reciprocal Harmonic Value
- What is a harmonic number? (Definition)
- How to calculate a harmonic number?
- What are the first values of the Harmonic Series?
- Is the Harmonic Series convergent?
- What is the relationship between harmonic numbers and the Riemann Zeta function?
- How to implement the Harmonic series?
