# Minkowski question mark

If $x \in \mathbb{R}$ is irrational with continued fraction expansion $x=[a_0;a_1,a_2,\ldots]$ then define $$?(x) = a_0 + 2 \displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}}{2^{a_1+\ldots+a_n}}.$$ If $x \in \mathbb{R}$ is rational with continued fraction expansion $x=[a_0;a_1,a_2,\ldots,a_m]$ then define $$?(x) = a_0 + 2\displaystyle\sum_{n=1}^m \dfrac{(-1)^{n+1}}{2^{a_1+\ldots+a_m}}.$$