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You can see the formula Piskorowski used here [0], under "Best Approximations to e with n Copies of the Digit k". (All formulas marked with "MP").

He did in fact build up (1 + x^(-1))^x to slowly increase the accuracy of the formula.

[0] https://www2.stetson.edu/~efriedma/mathmagic/0804.html (The page may say 2004, but has been actively updated, and has references to 2019.)




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