Inspired by this question, the series $\dfrac{1}{2^{2^{0}}}+\dfrac{1}{2^{2^{1}}}+\dfrac{1}{2^{2^{2}}}+\dfrac{1}{2^{2^{3}}}+\dots$ is clearly irrational.
But is it algebraic or transcendental?
I was thinking of answering this question by checking whether or not it can be represented as a periodic continued fraction:
- If no, then (as far as I know) it is transcendental
- If yes, then (as far as I know) we cannot infer the answer
But how do I determine whether or not it can be represented as a periodic continued fraction?
Is there a better way for tackling this question, or is the answer already known by any chance?
UPDATE:
Based on @Wojowu's comment:
- If it can be represented as a periodic continued fraction, then it is algebraic
- If it cannot be represented as a periodic continued fraction, then we cannot infer the answer
It is a general theorem proven by Mahler that given an integer $d>1$ and a nonzero algebraic number $z\in(-1,1)$ then the sum of the series $\sum_{n=0}^\infty z^{d^n}$ is a transcendental number. I can't find a direct reference, but you can find this theorem in this MO answer.
Transcendence of the number in your question follows by taking $d=2,z=\frac{1}{2}$.