This is going to be a long question; please bear with me.
We are familiar with the notations $\Sigma^{n}_{k=0} a_k$ and $\Pi^{n}_{k=0}a_k$ for the sum and product of the finite sequence $\{a_n\}$. I've recently been looking into whether we could extend this to exponentiation.
[In this context, we will assume $a^{b^c}$ = $a^{(b^c)}$.] We define $$\Delta^n_{k=1}a_k = \large{a_1^{a_2^{a_3^{^\cdots}}}}$$ Hence we see that $$\Delta^n_{k=1}a_k = a_1^{\LARGE{\Delta^n_{k=2}a_k}}$$ I've worked out derivatives etc. for this. It (this notation, or this way of looking at it) also seems to be helpful in finding the derivative of $a$ tetrated to $b$ (analogous to $\frac{d}{dx}f(x)^{g(x)}$).
My question is, what does this limit evaluate to? (Does it even exist?) $$\lim_{x\rightarrow\infty}\Large{\Delta^x_{k=1}}\normalsize{\sin \frac{k\pi}{2x}}$$
In the spirit of Math.SE, hints are all I ask for. I feel (probably mistakenly) that this delta thing is something important I've stumbled upon :)
after writing a quick script i can tell you that it evaluates to 1. i don't know how to proof it though.
edit: think i made a mistake, will post after i checked something
edit2: http://pastebin.com/fsSbj83k
also, this actually kind of makes sense. every individual term is between 0 and 1. something between 0 and 1 to the power something between zero and 1 will get closer to 1.