Translate the continuous fraction $[2;2^{1!}, 2^{2!}, 2^{3!}, \cdots]$ to a sum in order to prove that it's transcendental
The task is to prove that the number $[2;2^{1!}, 2^{2!}, 2^{3!}, \cdots]$ is transcendental.
The usual way is to show that it's a Liouville number, which will conclude that it's transcendental.
However, I know how to do it to numbers in the form of $\sum_{n=0}^{\infty} \frac{1}{2^{n!}}$ for example, and here I am struggling to bring the number $[2;2^{1!}, 2^{2!}, 2^{3!}, \cdots]$ to a form of a sum.
So, I simply need help in turning it to a form of a sum.