Factorial series
= Factorial series
{title2=$\sum_{n\geq1}a_n/n!$}
A factorial series with digits $0\leq a_n<n$ is an absolutely convergent <series> of <real numbers> between zero and one. The <telescoping series> identity
$$
\frac{n-1}{n!}=\frac1{(n-1)!}-\frac1{n!}
$$
shows 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>.