If
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
is a positive rational number show that
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
can be uniquely expressed in the form
![\displaystyle x = \sum^n_{k=1} \frac{a_k}{k!}](/media/m/b/b/7/bb73039f43b1cbc4f8684f1f7af81341.png)
where
![a_1, a_2, \ldots](/media/m/7/0/a/70a1247a1ab99045cbfc0d28aca8e204.png)
are integers,
![0 \leq a_n \leq n - 1](/media/m/5/6/b/56b3d75a576a890e84e6f4428f9fcc73.png)
, for
![n > 1,](/media/m/a/5/5/a5594d91e9be511e99e3bbed777bbe12.png)
and the series terminates. Show that
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
can be expressed as the sum of reciprocals of different integers, each of which is greater than
%V0
If $x$ is a positive rational number show that $x$ can be uniquely expressed in the form $\displaystyle x = \sum^n_{k=1} \frac{a_k}{k!}$ where $a_1, a_2, \ldots$ are integers, $0 \leq a_n \leq n - 1$, for $n > 1,$ and the series terminates. Show that $x$ can be expressed as the sum of reciprocals of different integers, each of which is greater than $10^6.$