cos cos cos ...

Note that, for any $x \in \mathbb{R}$ , we have $\cos(x) \in [-1,1]$ and hence $\cos(\cos(x))=[\cos(1),1]\approx [0.54,1]$ . Denote the compact set $[\cos(1),1]$ by S. Since the function $f(\cdot)=\cos(\cdot)$ is a non-increasing function in the interval $S$ , it is evident that $f$ maps all points in $S$ onto $S$ . Now consider the mapping $f :S\to S$ . For any $x \in S, |f'(x)|=\sin(x)\leq \sin(1) < 1$ . Hence the mapping $f:S\to S$ is a contraction . Now from a direct application of Banach's fixed point theorem , we conclude that the limit in the question exists and is equal to the unique fixed point of the equation $\cos(x)=x$ in the set $S$ , which is approximately $0.739$ .

This argument can be made more rigorous but here is the general point.

Let $x=(f\circ f \circ f \circ \cdots)(t)$ be the value denoting an infinite convolution of the function $f$ evaluated at $t$ . Applying $f$ to both sides preserves the RHS and gives us $f(x)=x$ as a functional equation.

Now for the case $f=\cos$ we are solving for $\cos(x)=x$ which is not doable explicitly so just grab a calculator and take cos of 1 50 times until the fifth decimal place stops changing or use a CAS. I don't know of anyway to solve such an equation so I will try and have a look around about it.

The answer is $\boxed{0.739}$ .

Similar solution posted here .

I tried to answer this sum logically instead of working it out.Now the range of cosine function is between -1 and 1.So if we arbitrarily choose a suitable angle like ( say 0 degrees or 60 degrees) we would see that the argument goes on diminishing and finally tends to 0. As we extrapulate n to infinity the argument would become 0 and as a result cosine of 0 degrees is equal to one and hence the result

So: Let: $S = \lim_{n \to \infty} (f \circ f \circ \ldots \circ f)$ Then:

$S(x) = y$ and $\cos(S(x)) = y$ So, substitute the first equation into the second:

$cos(y) = y$

Using like Taylor Series ( $x = \cos(x)$ has no closed form), or better yet just Wolfram Alpha for a value, you get:

$0.99985$

(remember to tell Wolfram Alpha degrees, or else you'll get 0.7 something, like I first did)

I hope I did this correctly.