I read on Wikipedia that
$$\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$$
to a high degree of accuracy. Why is this true? Is this pure coincidence or is there some mathematical background?
I read on Wikipedia that
$$\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$$
to a high degree of accuracy. Why is this true? Is this pure coincidence or is there some mathematical background?
It is a well known coincidence that
$$e^{\pi}-\pi \approx 20$$
Using this, we find
$$e^{\pi}-\pi \approx 20 \implies \pi\approx \log ( 20+\pi)$$
then
$$-1 =\cos (\pi) \approx \cos(\log ( 20+\pi))$$
$\cos (-\pi)=-1$, so a closer approximation of $-1$ can be found with
$$-1 =\cos(\pi\cos (\pi)) \approx \cos(\pi\cos(\log ( 20+\pi)))$$ and again
$$-1 =\cos(\pi \cos(\pi\cos (\pi))) \approx \cos(\pi\cos(\pi\cos(\log ( 20+\pi))))$$
In fact, if $x_0 \approx -1$ and $x_n=\cos (\pi x_{n-1})$ then $$\lim_{n \to \infty}x_n=-1$$