How would I prove (or disprove) that these following numbers are Liouville:
$\displaystyle\sum^\infty_{n=0}\frac{1}{\left(n!\right)!}$
$\displaystyle\sum^\infty_{n=0}\left(n!\right)^{-n!}$
Both of these numbers seem to have continued fractions that have mirroring* terms and terms that grow larger and larger very rapidly, without bound.
And I'm stuck. I know what makes a Liouville number what it is, but still, I'm stuck.
Usually in these cases you should work in this way: $$A_n=\sum_{i=0}^n a_i$$ Since $a_n\to 0$ very fast and $A_n$ is a rational number you may hope that $\forall n$ large enough $$\forall c>0,\ \ \sum_{i=n}^\infty a_i<(den(A_n))^{-c}$$ For example, in the first case $$den(A_n)=(n!)!$$ while for $n$ large enough $$\sum_{i=n}^\infty a_i<\frac{2}{((n+1)!)!}$$ since $$\frac{2(n!)!^c}{((n+1)!)!}\to 0$$ for all $c$ you have done