Loosely speaken, Liouville's theorem shows that rational series converging "too fast", have a transcendental limit. The concrete criterion is somewhat cumbersome and hard to check.
Now my question :
Is
$$\sum_{k=1}^\infty {\frac{1}{k \uparrow \uparrow n}}$$
a transcendental number for all natural numbers n>=2 ?
Is Liouvilles criterion helpful here ?
And finally, is the sum
$$\sum_{k=1}^\infty {\frac{1}{k \uparrow \uparrow k}}$$
a transcendental number ?