The n-th harmonic number, denoted as $H_n$, is the sum of the reciprocals of the first n positive integers.

The Formula:

$$ H_n = \sum_{k=1}^{n} \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} $$

The harmonic series (the sum as n goes to infinity) diverges, but it does so very slowly.

Approximation:

For large n, the harmonic number can be approximated using the natural logarithm and the Euler–Mascheroni constant ($\gamma \approx 0.57721$).

$$ H_n \approx \ln(n) + \gamma + \frac{1}{2n} $$