Just a quick reminder: you absolutly can calculate the ratio by 1 + sqrt(2) / 5 just as well as with your "feed back loop". Two methods for the same results.
Here your "feed back loop" formula is shown under the section "continued form" and the square root formula is shown under the "algebraic form" section.
BTW, fun fact: How much decimals of PI does NASA use for their highest precision orbital calculations? 10,000? 5,000? 1,000?
Just 15! For the Voyager spacecraft this imprecision equates with an error of its position of only 1.5 inches after travelling more than 12.5 billion miles.