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### Need help solving Recursive series defined by $x_1 = \sin x_0$ and $x_{n+1} = \sin x_n$ [duplicate]

$x_1 = \sin x_0 > 0$ $x_{n+1} = \sin x_n$ Prove $\lim_{x \to \infty }$ $\sqrt{\frac{n}{3}}$ $x_n = 1$ having problem of trying to figure out what value for the $x_0$ starts at.
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1 vote
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### Determine the convergence of the recursion $x_n=\sin(x_{n-1})$ [duplicate]

I want to determine the convergence of $$\begin{cases}x_0=1 & \\ x_n=\sin(x_{n-1}) \end{cases}$$ I can see that $x_n=\sin(x_{n-1}) \geq -1$. Which means the sequence is bounded. However, it isn'...
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### $x_1=\sin(x_0)>0, x_{n+1}=\sin(x_n)$, prove $\sqrt{\frac{n}{3}}x_n \to 1$ as $n \to \infty$ [duplicate]

I'm studying convergent sequences at the moment. And I came across this question in the section of Stolz Theorem. I realised that $\{x_n\}$ is monotonously decreasing and has a lower bound of $0$, ...
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### How fast does $f_n = \sin f_{n-1}$ approach zero? [duplicate]

The sequence $f(n) = \sin(\sin(\sin(......(1)......)))$ approaches zero like $\sqrt{3/n}$, as has been asked and answered here a few times. So $f(n)$ would get below $1/n$ after $3n^2$ steps, but it ...
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### How do I solve this limit: $\lim_{x \to \infty } \sqrt{n}\sin(\sin(\sin ... (\sin (1))...))$ [duplicate]

I have been strugling a lot to solve this question, but couldn't figure out where to start. $$\lim_{x \to \infty } \sqrt{n} . \underbrace {\sin(\sin(\sin ... (\sin (1))...))}_{n...times..}$$ I ...
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### Calculating $\lim_{n\to\infty}\sqrt{n}\sin(\sin...(\sin(x)..)$
I was asked today by a friend to calculate a limit and I am having trouble with the question. Denote $\sin_{1}:=\sin$ and for $n>1$ define $\sin_{n}=\sin(\sin_{n-1})$. Calculate \$\lim_{n\to\infty}\...