Liouville constant
= Liouville constant
{c}
{title2=$L=\sum_{n\ge1}10^{-n!}$}
The <Liouville constant> is the <real number>
$$
L=\sum_{n=1}^{\infty}10^{-n!}=0.110001000000000000000001\ldots.
$$
Its decimal ones occur at factorial positions $1,2,6,24,\ldots$. The growing gaps make $L$ exceptionally well approximated by <rational numbers>. If $p_N/q_N$ is the truncation after $N$ terms, with $q_N=10^{N!}$, then
$$
0<L-\frac{p_N}{q_N}<2\,10^{-(N+1)!}=\frac2{q_N^{N+1}}.
$$
The strict lower bound shows the truncations never equal $L$.
For any fixed <degree of an algebraic number> $d$, these errors eventually fall below every positive constant times $q_N^{-d}$. The <Liouville approximation theorem> forbids this for an irrational <algebraic number> of degree $d$. A rational number $a/b$ is also impossible, since distinct fractions satisfy $|a/b-p_N/q_N|\ge1/(bq_N)$. Hence $L$ is a <transcendental number>. The construction turns deliberately sparse decimal digits into a proof of transcendence.