Find the limit of sequence

**radian92** · November 12th 2011, 09:32 AM

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?

**girdav** · November 12th 2011, 11:56 AM

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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