Simple high-school maths.

    S = 0.00010002000400080016...

    S = 0.0001 + 0.0000 0002 + 0.0000 0000 0004 + 0.0000 0000 0000 0008 + ...

    S = 2^0 / 10000^0 + 2^1 / 10000^1 + 2^2 / 10000^2 + 2^3 / 10000^3 + ...

    S = sum to infinity of (2/10000)^i

You might have noticed this is a geometric series with ratio 2/10000 = 0.0002.

    S = 0.0001 / (1 - 0.0002) = 0.0001 / 0.9998 = 1/9998