If I put my calculator into radian mode, enter any number into it, and press the $cos$ key until my thumb is sore, the output tends to $0.739085133$? So $$\cos \cos \cos \cos \cos \cos (x) \to 0.739085133 $$
A few questions:
1. What's this behaviour called?
2. How do I express it correctly?
3. Does this number have any other expression, like $\frac{\pi^2}{6}$? (It's not that, but I hope you get the point.)
On
As you wrote, this number is the zero of function $$f(x)=x-\cos(x)$$ If you apply Newton method starting with a guess $x_0$, the iterates will be given by $$x_{n+1}=\frac{x_n \sin (x_n)+\cos (x_n)}{\sin (x_n)+1}$$ and then Dottie number is the limit of $x_n$ when $n \to \infty$.
Starting iterations using $x_0=1$, Newton iterates would be $$\left( \begin{array}{cc} n & x_n \\ 0 & 1.000000000000000000000000000000000000000000000000 \\ 1 & 0.750363867840243893034942306682176853246993065855 \\ 2 & 0.739112890911361670360585290904890234002892836736 \\ 3 & 0.739085133385283969760125120856804332889533123170 \\ 4 & 0.739085133215160641661702625685026372325223262530 \\ 5 & 0.739085133215160641655312087673873404013420776367 \\ 6 & 0.739085133215160641655312087673873404013411758901 \end{array} \right)$$
This is a fixed point of the cosine function: it is a solution to the equation
$$ \cos(x) = x $$
It is, in fact, the only fixed point, and it is a very good attractor, since no matter what number you start with, repeatedly applying cosine will give a sequence converging to this fixed point.