Evaluate $\frac{2}{4}\frac{2+\sqrt{2}}{4}\frac{2+\sqrt{2+\sqrt{2}}}{4}\cdots$ Evaluate 
$$
\frac{2}{4}\frac{2+\sqrt{2}}{4}\frac{2+\sqrt{2+\sqrt{2}}}{4}\frac{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}{4}\cdots .
$$
First, it is clear that terms tend to $1$. 
It seems that the infinity product is not 0. This is related to the post Sequence $x_{n+1}=\sqrt{x_n+a(a+1)}$.

Consider the corresponding finite product containing $n+1$ factors. Multiplying the last  factor (with $n$ square roots) by
$$2-\underbrace{\sqrt{2+\sqrt{2+\ldots}}}_{x_n},$$
the product telescopes to $4^{-n}$. 
On the other hand, since
$x_n^2-2=x_{n-1}$,
writing $x_n=2\cos\varphi_n$ we get $2\varphi_n=\varphi_{n-1}$, and therefore $\varphi_n=\frac{\varphi_1}{2^{n-1}}=\frac{\pi}{2^{n+1}}$. This allows to make an estimate
$$2-x_n=2-2\cos\varphi_n\approx \frac{\pi^2}{4}4^{-n},$$
which finally gives the answer:
$$\boxed{\displaystyle\lim =\frac{4}{\pi^2}}$$
