Given function $$f_n(x) = \cos x - (\cos \cos x) + (\cos \cos \cos x) - (\cos \cos \cos \cos x) + \dots + (-1)^{n-1} \underbrace{ \cos \cos \dots \cos }_n x,$$ where $n \in \mathbb{N}$ and $\underbrace{ \cos \cos \dots \cos }_n$ means cosine of cosine of cosine and so on $n$ times, find value of $$\sup_{n \rightarrow \infty, x \in \mathbb{R}} \{f_n(x)\}^{n}_{k=1}$$

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    $\begingroup$ Probably won't be much help but it seems that as n gets large the last terms will send to a maximum of around 0.74. $\endgroup$ – Michael Stachowsky Oct 9 '18 at 0:02
  • $\begingroup$ The value 0.74 is close to the root of $\cos x = x$. $\endgroup$ – Sungjin Kim Oct 10 '18 at 4:48
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    $\begingroup$ I think that this is deeply related to recurrence relations since we can define the problem as the evaluation of the series $\sum_{n\in\mathbb{N}} \left(\left(-1\right)^{n-1}{c_n\left(x\right)}\right)$ where $c_0(x)=x$ and $c_n(x)=\cos\left(c_{n-1}(x)\right)$. Adding the tag can give this post most more visibility to the recurrence relations experts $\endgroup$ – Fabio Jan 10 '19 at 14:13
  • $\begingroup$ The supreme is about 1.5708. It is heavily depend on the value of the first term, so we take $x=0$, then we can get the supreme. $\endgroup$ – mathon Sep 25 '19 at 9:15

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