标题： 柯里神秘的模式和迭代函数 显示英文标题

标题： Currie's Mysterious Pattern and Iterated Functions

摘要： 在他的著作《数学韵律与理性》中，Currie讨论了他所谓的涉及序列$mysterious$ $pattern$ 的$ a_{n} = 2^n \sqrt{2 - \sqrt{2 + \sqrt{2 + \cdots + \sqrt{2}}}},$，其中$n$是根的数量。 谜团的一部分在于$a_n$收敛到$\pi.$。在本文中，我们讨论了在迭代函数背景下类似神秘模式的结果的通用框架。 显示英文摘要

摘要： In his book "Mathematics Rhyme and Reason," Currie discusses what he calls a $mysterious$ $pattern$ involving the sequence $ a_{n} = 2^n \sqrt{2 - \sqrt{2 + \sqrt{2 + \cdots + \sqrt{2}}}},$ where $n$ is the number of radicals. Part of the mystery is that $a_n$ converges to $\pi.$ In this paper we discuss a general framework for results like the mysterious pattern in the context of iterated functions.