= Irrationality criterion for factorial series
{title2=$\sum a_n/n!\notin\mathbb Q$}
A <factorial series> with integral digits $0\leq a_n<n$ is an <irrational number> exactly when positive digits occur infinitely often and digits below their maximum occur infinitely often. Under these two conditions every scaled tail $N!\sum_{n>N}a_n/n!$ lies strictly between zero and one. If the sum were a <rational number> $A/B$, choose $N\geq B$; then $N!A/B$ and the scaled finite head are <integers>, contradicting the strict tail bound. Conversely, an eventually zero digit sequence gives a finite <rational number> sum, and an eventually maximal sequence gives a finite head plus $1/N!$, again a <rational number>.
Back to article page
