Let $\displaystyle a_0=1, a_{n+1}=sin(a_n)$.

Does the constant $\displaystyle \alpha$ which $\displaystyle n^{\alpha}a_n$ has positive and finite limit exist?

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- Nov 12th 2011, 09:32 AMradian92Find the limit of sequence
Let $\displaystyle a_0=1, a_{n+1}=sin(a_n)$.

Does the constant $\displaystyle \alpha$ which $\displaystyle n^{\alpha}a_n$ has positive and finite limit exist? - Nov 12th 2011, 11:56 AMgirdavRe: Find the limit of sequence
Yes: you can show that $\displaystyle \lim_{n\to\infty}\frac 1{a_{n+1}^2}-\frac 1{a_n^2}$ is a positive constant (I found $\displaystyle \frac 13$, but it doesn't matter). Therefore, you can show that $\displaystyle \lim_{n\to\infty}\frac 1{na_n^2}$ is this constant, hence $\displaystyle a_n\sim \frac{\sqrt 3}{\sqrt n}$.