Find the value of the infinite continued fraction

$$
1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{\ldots}}}} .
$$
Let the value of the fraction be $x$. Then we get

$$
x=1+\frac{1}{x+1} \Longrightarrow x(x+1)=(x+1)+1 \Longrightarrow x^{2}=2
$$

Thus, since $x$ is positive, we must have $x=\sqrt{2}$.