Factorial series (${\sum }_{n\ge 1}{a}_n/n!$)

A factorial series with digits $0\le a_n<n$ is an absolutely convergent series of real numbers between zero and one. The telescoping series identityshows that its largest possible tail after place $N$ is $1/N!$. Multiplying this tail by $N!$ gives a number between zero and one, a useful arithmetic constraint on rational numbers.

A factorial series with integral digits $0\le a_n<n$ 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 $N\ge B$; 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.