How to compute Dottie number accurately? Dottie number is root of this equation : $cos \alpha = \alpha$, $\alpha \approx 0.73908513321516064165531208767\dots$.
I wonder how can I compute it ? I have tried to do it with an approximating formula:
$\alpha = \frac{5\pi^2}{\alpha^2 + \pi^2} - 4$
I have solved this equation and i got $\alpha \approx 0.738305\dots$. So , how can i compute it accurately ? Can i use taylor series, etc. ?

Let $a_{0}$ be the start input (it's better to choose one close to Dottie's Number).
For example, I'll choose it as $a_{0} = 0.73$.
$$a_{n} = \cos(a_{n-1})$$
$$ \lim_{n \to \infty} a_{n} = \text{Dottie's Number}$$
Let's begin the calculations:
$$a_{1} = \cos(a_{0}) = \cos(0.73) = 0.745174402...$$
$$a_{2} = \cos(a_{1}) = \cos(0.745174402...) = 0.734969653...$$
$$a_{3} = \cos(a_{2}) = \cos(0.734969653...) = 0.741851103...$$
$$a_{4} = \cos(a_{3}) = \cos(0.741851103...) = 0.737219118...$$
$$\dots$$
$$a_{55} = 0,739085133...$$
Of course it is not easy to do by hand, but for a computer it's a piece of cake.
