# Is there an analytic solution for such problem?

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 $$\sup_{n \rightarrow \infty, x \in \mathbb{R}} \{f_n(x)\}^{n}_{k=1}$$

• 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. – Michael Stachowsky Oct 9 '18 at 0:02
• The value 0.74 is close to the root of $\cos x = x$. – i707107 Oct 10 '18 at 4:48
• 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 – Fabio Jan 10 at 14:13