Repeated applications of cos function converges around $0.7390851$

Question

I set my calculator to radian mode. I enter any random number $x$. Then I repeatedly apply cosine to that number, getting the sequence $$\cos x, \cos(\cos x), \cos(\cos(\cos x)) \ldots$$ and so on. No matter what $x$ I start with, it always converges around $x^* = 0.7390851$. Obviously, $x^*$ is the solution of $\cos x=x$. But why does repeated application of $\cos$ converge around this number?

Answer 1

There are a couple of possibilities:

1. The sequence will converge.
2. The sequence will diverge.
3. The sequence will cycle.

You can check that if $x$ is a fixed point of $f$ and $|f'(x)|<1$ then the fixed point is stable (i.e., the sequence will converge). Validate this for your particular problem. More details can be found at this link.