<s>[INST] Why is $\frac{987654321}{123456789} = 8.0000000729?!$ Many years ago, 
I noticed that $987654321/123456789 = 8.0000000729\ldots$.
I sent it in to Martin Gardner at Scientific American
and he published it in his column!!!
My life has gone downhill since then:)
My questions are:


*

*Why is this so?

*What happens beyond the "$729$"?

*What happens in bases other than $10$?
 [/INST] In base $n$ the numerator is $$p = n^{n-1} - \frac{n^{n-1}-1}{(n-1)^2}$$ and the denominator is $$q = \frac{n(n^{n-1}-1)}{(n-1)^2}-1.$$
Note that $p = (n-2)q + n-1$ and for the quotient we get
\begin{align}
\frac{p}{q} &= n-2 + \frac{(n-1)^3}{n^n} \frac{1}{1 - \frac{n^2-n+1}{n^n}} \\
&= n-2 + \frac{(n-1)^3}{n^n} \sum_{k=0}^{\infty} \left(\frac{n^2-n+1}{n^n}\right)^k.
\end{align}
Indeed for $n=10$ this is
$$\frac{987654321}{123456789} = 8 + \frac{729}{10^{10}}\sum_{k=0}^{\infty}\left(\frac{91}{10^{10}}\right)^k
$$
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