Convergence of a sequence with repeated sines

Question

Let $x\in(0,\pi/2)$ and $\{a_n\}_{n\in\mathbb N}$ defined recursively as follows: $$ a_0=x, \quad \text{and} \quad a_{n+1}=\sin(a_n). $$ Show that $$ \lim_{n\to\infty}{n\,a_n^2}=3. $$

Note. There is a very strong numerical evidence that the above is valid, and the limit is independent of the initial choice of $a_0$. I have no idea where to start from.

You probably want to exclude $\pi$ from the allowed starting values ;) — Daniel Fischer, Dec 10 '13 at 22:21
You are absolutely right. I meant $x\in(0,\pi/2)$! It is now corrected. — Yiorgos S. Smyrlis, Dec 10 '13 at 22:26

score 0 · Answer 1 · answered Dec 10 '13 at 22:22

0

The asymptotics of $a_n$ are completely analyzed in de Bruijn's book, p. 157, from which this follows immediately.

answered Dec 10 '13 at 22:22

Igor Rivin

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Convergence of a sequence with repeated sines

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