Problem 911

Khinchin Exceptions

An irrational number $x$ can be uniquely expressed as a continued fraction $[a_0; a_1,a_2,a_3,\dots]$:
$$x=a_{0}+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+{_\ddots}}}}$$
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, $k_j(x)=(a_1a_2 \cdots a_j)^{1/j}$.

Also define $k_\infty(x)=\lim_{j\to \infty} k_j(x)$.

Khinchin proved that almost all irrational numbers $x$ have the same value of $k_\infty(x)\approx2.685452\dots$ known as Khinchin’s constant. However, there are some exceptions to this rule.

For $n\geq 0$ define
$$\rho_n = \sum_{i=0}^{\infty} \frac{2^n}{2^{2^i}}$$
For example $\rho_2$, with continued fraction beginning $[3; 3, 1, 3, 4, 3, 1, 3,\dots]$, has $k_\infty(\rho_2)\approx2.059767$.

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

辛钦例外

任意无理数$x$可以被唯一地表示为连分数$[a_0; a_1,a_2,a_3,\dots]$》
$$x=a_{0}+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+{_\ddots}}}}$$
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，$k_j(x)=(a_1a_2 \cdots a_j)^{1/j}$。

同时定义$k_\infty(x)=\lim_{j\to \infty} k_j(x)$。

辛钦证明了，几乎所有无理数$x$对应的$k_\infty(x)$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$n\geq 0$，定义
$$\rho_n = \sum_{i=0}^{\infty} \frac{2^n}{2^{2^i}}$$
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$k_\infty(\rho_2)\approx2.059767$。

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