harmonic number

Definition

The $n$-th harmonic number is:

$$H_n=\sum _{k=1}^n\frac{1}{k}$$

Useful Bounds

$$\ln (n+1)\le H_n\le \ln (n)+1$$

Estimation

For large $n$:

$$H_n\approx \ln (n)+\gamma$$

• $\ln (n)$: natural logarithm
• $\gamma$: Euler–Mascheroni constant $\approx 0.57721$

Refined Estimate

$$H_n=\ln (n)+\gamma +\frac{1}{2n}-O\left(\frac{1}{n^2}\right)$$