100/89= 1.1235..., which includes the first five Fibonacci numbers.

10000/9899=1.010203050813213455... includes the first ten Fibonacci numbers.

1000000/998999=1.001002003005... produces the first 15 Fibonacci numbers.

If you add two zeros in the numerator and two nines (one at the beginning, one at the end) in the denominator, does this Fibonacci production go on?

The fractions are 10^2k/(10^2K-10^K-1), so writing z=10^K, we have
z^2/(z^2-z-1). Finally, if we let x=1/z, we get f(x)=1/(1-x-x^2), which
is a generating function for the Fibonacci numbers: f(x)=ΣF

_{n}x^n, and since x=10^-K, the result follows.