A simple way to figure out how this works is to figure out another way to write it out. For the simpler case (1/9801) = 0.00010203... 0.00 + 0.0001

A simple way to figure out how this works is to figure out another way to write it out. For the simpler case (1/9801) = 0.00010203... 0.00 + 0.0001 + 0.000002 + 0.00000003 ... -------------- Each row is equal to x, but shifted over 2x digits. This is the same as dividing by 10^x. This simplifies to the formula: sum k=0 to infinity: k/(10^k) This is fairly easily calculable, and results in 1/9801. Try it yourself on wolfram alpha: http://www.wolframalpha.com/input/?i=%28sum_%28k%3D1%29%5Einfinity%28%28k-1%29%2F%28100%5Ek%29%29 http://www.wolframalpha.com/input/?i=%28sum_%28k%3D1%29%5Ein...