Liouville constant ($L={\sum }_{n\ge 1}10^{-n!}$)

The Liouville constant is the real numberIts decimal ones occur at factorial positions $1,2,6,24,\dots$. The growing gaps make $L$ exceptionally well approximated by rational numbers. If $p_N/q_N$ is the truncation after $N$ terms, with $q_N=10^{N!}$, thenThe 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 $|a/b-p_N/q_N|\ge 1/(b{q}_N)$. Hence $L$ is a transcendental number. The construction turns deliberately sparse decimal digits into a proof of transcendence.