Irrationality of e
= Irrationality of e
{title2=$e\notin\mathbb Q$}
If <Euler's number> were $a/b$, choose $n\geq\max(b,2)$ and multiply its factorial-series tail by $n!$. This produces an <integer>, but the tail is positive and smaller than $\sum_{j\geq1}(n+1)^{-j}=1/n<1$. The contradiction proves that the number is an <irrational number>. The argument illustrates how a rapidly convergent rational series can <force> a hypothetical rational remainder into an impossible <integer> interval.