Put $a_{0}= \pi/2$ and $a_{n}=\sin(a_{n-1})$.
Is $\sum a_{n}$ a convergent series?
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Siminore
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Ali Taghavi
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1What are your thoughts? – Siminore Apr 10 '17 at 07:52
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@Siminore Thank you for your edit. I have no idea for this series. But this question can introduce the following concept in non hyperbolic fixed points of maps (Then can be adapted for flows): WLOG we may assume that the fixed point of a function $f$ is $0$. We say $0$ is a conditionally fixed point if for every $p$ in a small neighborhood of $0$ $\sum f^{n} (p)$ converges. Of course this is the case for hyperbolic attractors.one can even think about the generalization for dynamics an arbitrary [https://en.wikipedia.org/wiki/Affine_manifold] (affine manifolds) – Ali Taghavi Apr 10 '17 at 08:37
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2Duplicate of https://artofproblemsolving.com/community/c7h1059961_does_sum_a_n_converge_if_a_n_sin_sin__sinx – Jack D'Aurizio Apr 10 '17 at 08:42
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3Possible duplicate of Does $\sum a_n$ converge if $a_n = \sin( \sin (...( \sin(x))...)$ – Claude Leibovici Apr 10 '17 at 08:43
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https://en.wikipedia.org/wiki/Affine_manifold – Ali Taghavi Apr 10 '17 at 08:43
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@JackD'Aurizio thank you for the link. – Ali Taghavi Apr 10 '17 at 08:44
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@ClaudeLeibovici thanks for the link. – Ali Taghavi Apr 10 '17 at 08:45