The pattern will break down once you get past 8192, which is 2^13. That means that the pattern continues for an impressive 52 significant figures (well, it actually breaks down on the 52nd digit, which will be a 3 instead of a 2).
The reason it works is that 9998 = 10^4 - 2. You can expand as
1 / (10^n - 2) = 1/10^n * 1/(1 - 2/10^n)
= 1/10^n * (1 2/10^n 2^2 /10^2n 2^3 /10^3n ...)
which gives the observed pattern. It breaks down when 2^k has more than n digi...
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