Consider $$\alpha = \sum_{k=0}^{\infty}2^{-(k+1+3^k)}$$
I think one can easily show using the Roth Siegel Thue theorem that $\alpha$ is transcendental because the irrationality exponent of $\alpha$ is almost $3$ and certainly above $2$. But since I have never dealt with these methods before I am seeking a confirmation here: is $\alpha$ above transcendental?
Many thanks in advance, Martin.