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Consider the following series generated by recursive function: $$a_{n+1}=\sin a_n$$

Apparently, $a_n$ is infinitesimal.

Apply Stolz Theorem to $\displaystyle\lim_{n\to\infty}\frac{n}{1/a_n}$, we can get: $$a_n\sim\sqrt{\frac{1}{3n}}$$

If we want to estimate it more accurately, such as: (This result is only for clearer description, not correct) $$a_n=\sqrt{\frac{1}{3n}}+\frac1n\sqrt{\frac{1}{n}}+o\left(\frac1n\sqrt{\frac{1}{n}}\right)$$

The step (Stolz) cannot continue.

What other methods can be used for this issue?

Alpha Beta
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  • What is $a_0$ ? – Claude Leibovici Aug 16 '23 at 09:45
  • See https://math.stackexchange.com/questions/3837335/proving-the-limiting-behavior-of-functions-containing-iterated-trigonometric-fun and any other of the many questions towards the "sine iteration". https://math.stackexchange.com/questions/1449281/iterated-sine-function-on-different-arguments looks nice too – Lutz Lehmann Aug 16 '23 at 10:56
  • The next terms in $\frac3{a_n^2}-\frac3{a_0^2}=n+...$ are constant and logarithmic in some complicated way, thus seldom explored. – Lutz Lehmann Aug 16 '23 at 11:31

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