Euler–Mascheroni constant

Description

The Euler–Mascheroni constant is the real number $$\gamma = \lim_{n \to \infty} \big( H_n - \ln n \big) \approx 0.5772156649,$$ where $H_n = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ is the $n$-th harmonic number.

Remarks

The limit exists, because the sequence $H_n - \ln n$ is decreasing and bounded below by $0$. Also $\gamma = \int_1^\infty \Big( \frac{1}{\lfloor x \rfloor} - \frac{1}{x} \Big) \, dx$. It is not known whether $\gamma$ is rational. Mathlib defines it as Real.eulerMascheroniConstant. Leonhard Euler introduced the constant in a paper of 1734. Lorenzo Mascheroni later computed it to more decimal places, with some errors.

Explanations

L1 human

The harmonic numbers grow like $\ln n$, and $\gamma$ is the limit of the difference. So $H_n \approx \ln n + 0.5772$ for large $n$.