What number does the recurrence $x_{n+1} = \cos(x_n) $ converge to?

Question

The recurrence $x_{n+1} = \cos(x_n) $ seems to converge to a value around .739 no matter what number is chosen for $x_0$, even complex numbers. What is the exact number that this recurrence converges to?

Answer 1

If the limit exists, then $L=\cos(L)$. The solution to this equation is the Dottie number.

Answer 2

Hints:

• $f: x\mapsto \cos(\cos x)$ is increasing on $[0,1]$.

• For all $n\ge 1$, $x_n \in [0,1]$

• Let $a_n = x_{2n}$ and $b_n = x_{2n+1}$. Use $f$ to prove that $(a_n)$ and $(b_n)$ both converge to the same value. Conclude that $(x_n)$ converges (the value to which it converges is the unique fixed point of $\cos$).