Harmonic Number

Question 1

Answer

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

Question 2

Answer

The first harmonic numbers are:

Nth Harmonic Number Calculator