1/9998 = 0.0001 0002 0004 0008 0016 0032 0064 0128 0256..

Breaks at 8,192 of course...

The fact that 10000 - 2 = 9998 probably has something to do with it. Compare and contrast the result for 1 / 99998.

To expend it all you have to do is add more nines.

Similarily, 1/9999 gives you powers of 1 (which is just 1), 1/9997 powers of 3, and so on.

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 digits, which happens approximately when

    2^k > 10^n   =>   k > n log(10) / log(2)

which comes out to 4 * log(10)/log(2) = 13.28 when n = 4.

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Another pattern can be generated from the power series expansion

    x / (1 - x)^2 = x + 2x^2 + 3x^3 + 4x^4 + ...

setting x = 1/10^n gives the infinite series

    1/10^n + 2/10^2n + 3/10^3n + ...

which leads to the neat fact that

    1 / 998001 = 0.000 001 002 003 004 005 006 007...

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Another example is the fraction

    1000 / 997002999 = 0.000 001 003 006 010 015 021 ...

which goes through the triangle numbers[0] in its expansion, or

    1 / 998999 = 0.000 001 001 002 003 005 008 013 021 ...

which goes through the Fibonacci numbers[1].

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Getting the squares is harder, but you can do it with

    1001000 / 997002999 = 0.001 004 009 016 025 036 049 ...

[0] http://en.wikipedia.org/wiki/Triangle_number

[1] http://en.wikipedia.org/wiki/Fibonacci_number