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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?

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    How much calculus do you know? Beyond the initial coincidence $\log (20 + \pi) \approx \pi$ the rest is more or less the Banach fixed point theorem (http://en.wikipedia.org/wiki/Banach_fixed-point_theorem) applied to $\cos \pi x$ near $x = -1$, but the easiest way to show this requires computing the derivative of $\cos \pi x$ at $x = -1$...2012-06-07

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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 $\lim_{n \to \infty}x_n=-1$