Harmonic Series

The harmonic series is the infinite series defined as the sum of the reciprocals of the natural numbers:

$$\sum _{n=1}^{\infty }\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$$

Convergence

The harmonic series diverges.

Approximation

The sum of the first $n$ terms of the harmonic series can be approximated using

$$\sum _{k=1}^n\approx \ln (n)+\gamma$$

for $n$ large, where $\gamma \approx 0.577$