26 - Euler's Constant

Euler's Constant

Let

$${\gamma }_n=1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}-\log n=\sum _{k=1}^n\frac{1}{k}-{\int }_1^n\frac{1}{x}dx$$

By Integral Test using $f:[1,\infty )\to \mathbb{R}$ by $f(x)=\frac{1}{x}$ then $({\gamma }_n)$ converges
Suppose

$${\gamma }_n\to \gamma \text{ as }n\to \infty$$

and $0\le \gamma \le 1$ (from the test)