Harmonic Number

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

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

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

The first harmonic numbers are:

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.

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

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
}