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Let $\left(a_{n}\right)_{n \geqslant 1}$ be the sequence defined by $$ \left\{\begin{array}{l} a_{1}=1 ; \\ \forall n \geqslant 1, a_{n+1}=\sin \left(a_{n}\right) \end{array}\right. $$ (a) Show that $\left(a_{n}\right)$ is decreasing and deduce that $\left(a_{n}\right)$ tends to $0$ .

(b) Show that there exists a unique real number $\alpha$ such that the sequence $\left(a_{n+1}^{\alpha}-a_{n}^{\alpha}\right)$ converges to some non-zero limit $\ell$. Give the values of $\alpha$ and $\ell$.

(c) Deduce that $a_{n} \sim \sqrt{\frac{3}{n}}$ as $n \rightarrow \infty$.

For question (a) I can do it. But for question (b) I cannot do it. Please kindly help me. THANK IN ADVANCED!

Peter Green
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  • From (a) both $(a_n)$ and $(\sin a_n)$ are null sequences. Thus the required $\alpha$ must be negative. Rewriting $\beta = -\alpha$ you get $a_{n+1}^\alpha - a_n^\alpha\sim \frac{a_n^\beta - \sin^\beta a_n}{a_n^{2\beta}}$. Using fundamental limits from here you get $\beta = 2$ and $\ell = \frac13$. – dfnu Jul 16 '21 at 20:54
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    This is in de Bruijn, Asymptotic Methods in Analysis, several pages on this example https://www.google.com/books/edition/Asymptotic_Methods_in_Analysis/7-wxAwAAQBAJ?hl=en&gbpv=1&printsec=frontcover – Will Jagy Jul 16 '21 at 21:10

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