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Math Help - limit of ln|sin(n)|/n, n->infinity

  1. #1
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    limit of ln|sin(n)|/n, n->infinity

    prove: \lim_{n}\cfrac {ln|sin(n)|} {n} = 0 , n\rightarrow \infty
    or show it is not true.
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    Re: limit of ln|sin(n)|/n, n->infinity

    I think the limit does not exist but i`m not sure.
    Try with L`hopital rule , you will get limn->inf from cot(n) , and that does not exist as the function is oscilating to infinity without aproaching any specific value.
    Sry for bad english
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  3. #3
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    Re: limit of ln|sin(n)|/n, n->infinity

    Since $\displaystyle \begin{align*} \sin{(x)} \end{align*}$ does not have an infinite limit, then I don't see how $\displaystyle \begin{align*} \ln{ \left| \sin{(x)} \right| } \end{align*}$ could possibly have one. So I would expect that the limit does not exist. Thus L'Hospital's Rule should not be used...
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    Re: limit of ln|sin(n)|/n, n->infinity

    0 \ge \lim_{n \to \infty} a_{n} \geq \limsup_{n \to \infty} a_{n} = 0
    Last edited by topsquark; August 16th 2014 at 05:05 PM. Reason: Fixed LaTeX
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  5. #5
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    Re: limit of ln|sin(n)|/n, n->infinity

    Quote Originally Posted by Cartesius24 View Post

    $ 0 \geq \displaystyle{\lim_{n \to \infty}} a_{n} \geq \displaystyle{\limsup_{n \to \infty}}~ a_{n} = 0 $
    corrected latex for our interpreter
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  6. #6
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    Re: limit of ln|sin(n)|/n, n->infinity

    Quote Originally Posted by Cartesius24 View Post
    0 \ge \lim_{n \to \infty} a_{n} \geq \limsup_{n \to \infty} a_{n} = 0
    May I ask what you got for the supremum of this sequence?
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    Re: limit of ln|sin(n)|/n, n->infinity

    I would try to show that for any fixed \varepsilon>0 there are infinitely many n's such that \sin x>1-\varepsilon and infinitely many n's such that |\sin x|<\varepsilon.

    The basic idea is to use Dirichlet principle.

    Similar approach was used here: Sine function dense in [−1,1] at MSE.
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    Re: limit of ln|sin(n)|/n, n->infinity

    Quote Originally Posted by Prove It View Post
    May I ask what you got for the supremum of this sequence?
    Doen't matter . It is 1. More important is this is finite.
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