Question 18
(GBR 5) If $x$ is a positive rational number, show that $x$ can be uniquely expressed in the form
$$x=a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\cdots,$$
where $a_1,a_2,\ldots$ are integers, $0\leq a_n\leq n-1$ for $n>1$, and the series terminates.
Show also that $x$ can be expressed as the sum of reciprocals of different integers, each of which is greater than $10^6$.
$$x=a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\cdots,$$
where $a_1,a_2,\ldots$ are integers, $0\leq a_n\leq n-1$ for $n>1$, and the series terminates.
Show also that $x$ can be expressed as the sum of reciprocals of different integers, each of which is greater than $10^6$.