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Math Help - Find the limit of sequence

  1. #1
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    Find the limit of sequence

    Let a_0=1, a_{n+1}=sin(a_n).
    Does the constant \alpha which n^{\alpha}a_n has positive and finite limit exist?
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  2. #2
    Super Member girdav's Avatar
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    Re: Find the limit of sequence

    Yes: you can show that \lim_{n\to\infty}\frac 1{a_{n+1}^2}-\frac 1{a_n^2} is a positive constant (I found \frac 13, but it doesn't matter). Therefore, you can show that \lim_{n\to\infty}\frac 1{na_n^2} is this constant, hence a_n\sim \frac{\sqrt 3}{\sqrt n}.
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