I was riding my bike yesterday and thinking about the following expression:
Suppose $$S_3 = \sin^2(\frac\pi2) + \sin^2(\sin^2(\frac\pi2)) + \sin^2(\sin^2(\sin^2(\frac\pi2)))$$ which serves to define an $$S_n$$ that has $n$ of these terms, the last of which being $n$ successive applications of $\sin^2$.
Now, you will appreciate that $$\lim\limits_{n \to \infty}{S_n}$$ goes "somewhere". I have no idea how to calculate it, and what type of maths would describe its calculation. All I know is that it seems to drift close to the religious rational number $\frac73$ - although as someone pointed out, not quite that number.
During the bike ride I was actually interested in building an expression $\lim\limits_{n \to \infty}(S_n-C)^{F(n)}$ and see if it combines $e$ and $\pi$ in a new expression. Note that for instance $$\lim\limits_{n \to \infty}{(1+\frac{1}{n^2})^{n^2}} = e$$ so even the quick "convergence" of $S_n$ can bloom open to something interesting when other operators are involved.
Now, sitting at my desk I realise that I don't even know where to start with that original idea, as I don't even know how to compute the limit.
Can anyone point me in a direction, does this iteration of $\sin^2$ converge? How does one prove it? Are there more things I could read about sine hyperoperators like these?
I'm probably overlooking something basic - I'm not that knowledgable in maths. Thanks for any hints!
If $x\le1$, $\sin^2(x)\le x\sin(1)\le1$ and the series is bounded by a converging geometric series of common ratio $\sin(1)$.
Below a plot of the first partial sums.