How close can an infinite decimal (with no zeroes except the leading one, and between 0 and 1) get to its continued fraction formed by using its digits after the decimal point? Example : difference between 0.3219… and continued fraction [0;3,2,1,9,…]
2026-05-10 17:13:34.1778433214
What is the minimum difference between decimal and continued fraction formed by its digits?
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It appears that as the number of digits increases, the optimal difference converges to around 0.02286, alternating above and below this limit. It looks like you have already noticed this.
Due to this alternating behaviour, the answer to your question is not so interesting, as it happens that the global minimum is near the start at [0,3,2,1] where the difference is just 0.021.