Limit of a sequence of fractions

Define $a_0(n) = \dfrac{2n-1}{2n}$ and $a_{k+1}(n) = \dfrac{a_{k}(n)}{a_{k}(n+2^k)}$ for $k \geq 0$ .

The first several terms in the sequences $a_k(1)$ for $k \geq 0$ are $\dfrac{1}{2}$ , $\dfrac{1/2}{3/4}$ , $\dfrac{\frac{1}{2}/\frac{3}{4}}{\frac{5}{6}/\frac{7}{8}}$ , $\dfrac{\frac{1/2}{3/4}/\frac{5/6}{7/8}}{\frac{9/10}{11/12}/\frac{13/14}{15/16}}$ , $\cdots$

What limit do the values of these fractions approach?

Use 5 significant digits.