Is there a nice explanation of
$$\frac{1}{998999}=0.000\,\underbrace{001}_{F_1}\,\underbrace{001}_{F_2}\,\underbrace{002}_{F_3}\,\underbrace{003}_{F_4}\,\underbrace{005}_{F_5}\,\underbrace{008}_{\ldots}\,013\,021\,034\,055\,089\,144\,233\,377\,…$$
or is this a mere coincidence?
The pattern breaks at $16$th Fibonacci number producing $988$ instead of $987$.
Best Answer
It is not a coincidence. Let $s(x)$ be the generating function of Fibonacci numbers. Then we have $$s(x)=\frac{x}{1-x-x^2}=F_0+F_1x+F_2x^2+\dots,\ \mbox{for}\ |x|<\frac{1}{\varphi},$$ where $\varphi$ is the golden ratio. Now put $x:=10^{-3}$ and you easily get your equality.