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I'm trying to find this limit:

$$\lim_{n \to \infty} \underbrace{\sin \sin \ldots \sin }_{\text{$n$ times}}x$$

Thank you

  • 2
    Please use LaTeX in the future, just as Jonas did for you on this question.2011-08-20
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    "Forgive me, father, for I have sinned. Many many times..."2011-08-20
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    Mark- Nice!! :) Raphael- I hate wearing latex..:) What is Latex?2011-08-20
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    @user14829 $\Large \textrm{Thi}\int_{s}^{i} \LaTeX.$2011-08-20
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    (I can make it `\HUGE` if you want. ;))2011-08-20
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    $\top \mathbb{N} \chi $2011-08-20
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    This question is not a duplicate. In the other question, $n$ and $x$ are the same.2011-08-23
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    @Qiaochu: But Aryabhata pointed to question 3215, which contains strictly the present one... (For the record, I was not aware of 3215 when I asked this question to be reopened, but I still think it should stay open.)2011-08-23
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    Banach's fixed theorem!?2011-08-23

2 Answers 2

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First, you have to prove there is a limit. the first application of $\sin x$ will get us into $[-1,1]$ If $x \in [0,1]$ we have $0 \le \sin x \lt x$, so the sequence (after the first term, maybe) is monotonically decreasing and bounded below by $0$. Then, if there is a limit, you must have $\sin x=x$. Where is that true? The case below zero is for you.

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Here is a different approach that doesn't lead as directly to a rigorous proof as Ross Millikan's, but is more concrete and shows the rate of convergence. Let the $n$th member of the sequence be given by the function $x(n)$. Using the Taylor series of the sine function, and approximating the discrete function $x$ by a continuous one, we have $dx/dn\approx -(1/6)x^3$. Separation of variables and integration gives $x\approx \pm\sqrt{3/n}$ for large $n$.

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    http://math.stackexchange.com/questions/3215/convergence-of-sqrtnx-n-where-x-n1-sinx-n/3220#32202011-08-23