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Problem 911

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Khinchin Exceptions

An irrational number $x$ can be uniquely expressed as a continued fraction $[a_0;a_1,a_2,a_3,\dots ]$:
$$x=a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+{}_{\ddots }}}}$$
where $a_0$ is an integer and $a_1,a_2,a_3,\dots$ are positive integers.

Define $k_j(x)$ to be the geometric mean of $a_1,a_2,\dots ,a_j$.

That is, $k_j(x)=(a_1{a}_2\cdots a_j{)}^{1/j}$.

Also define $k_{\mathrm{\infty }}(x)=\underset{j\to \mathrm{\infty }}{lim}{k}_j(x)$.

Khinchin proved that almost all irrational numbers $x$ have the same value of $k_{\mathrm{\infty }}(x)\approx 2.685452\dots$ known as Khinchin’s constant. However, there are some exceptions to this rule.

For $n\ge 0$ define
$${\rho }_n=\sum _{i=0}^{\mathrm{\infty }}\frac{2^n}{2^{2^i}}$$
For example ${\rho }_2$, with continued fraction beginning $[3;3,1,3,4,3,1,3,\dots ]$, has $k_{\mathrm{\infty }}({\rho }_2)\approx 2.059767$.

Find the geometric mean of $k_{\mathrm{\infty }}({\rho }_n)$ for $0\le n\le 50$, giving your answer rounded to six digits after the decimal point.

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辛钦例外

任意无理数$x$可以被唯一地表示为连分数$[a_0;a_1,a_2,a_3,\dots ]$》
$$x=a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+{}_{\ddots }}}}$$
其中$a_0$是整数，$a_1,a_2,a_3,\dots$都是正整数。

定义$k_j(x)$为$a_1,a_2,\dots ,a_j$的几何平均值。
换言之，$k_j(x)=(a_1{a}_2\cdots a_j{)}^{1/j}$。

同时定义$k_{\mathrm{\infty }}(x)=\underset{j\to \mathrm{\infty }}{lim}{k}_j(x)$。

辛钦证明了，几乎所有无理数$x$对应的$k_{\mathrm{\infty }}(x)$值都相同，约等于$2.685452\dots$，这个值被称为辛钦常数。然而，这个规则存在一些例外。

对于$n\ge 0$，定义
$${\rho }_n=\sum _{i=0}^{\mathrm{\infty }}\frac{2^n}{2^{2^i}}$$
例如，${\rho }_2$的连分数表示的初始部分是$[3;3,1,3,4,3,1,3,\dots ]$，对应的$k_{\mathrm{\infty }}({\rho }_2)\approx 2.059767$。

求$0\le n\le 50$范围内所有$k_{\mathrm{\infty }}({\rho }_n)$的几何平均值，并将你的答案四舍五入保留小数点后六位。

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