[kznllajg] Feynman point in Continued fraction?

Does the Feynman point appear in the continued fraction representation of pi?  That is, is the rational number that results if the six 9s were extended forever one of the convergents of the continued fraction?

This should be easy.

Update: The answer is No. Observing that the digit before the Feynman point is 4999999, both the numerator and the denominator are divisible by 5, so the approximation is 6.28e760/2e760, and the difference from pi is 1.627e-768.

However, the 1490th convergent has smaller denominator 1.8e760, and so the approximation is 5.8e760/1.8e760, and the difference from pi is 1.3e-1521, so correct to almost twice as many digits. This matches intuition, as the Feynman point approximation only has entropy in the numerator, while the continued fraction approximation has entropy in the numerator and denominator. Hand wavingly: correct to 753 more digits, plus 6 nines + 1 = 760 the size of the denominator.

contfrac :: [Integer] -> Rational ; contfrac [] = 0 ; contfrac [x] = fromInteger x ; contfrac (h:t) = (fromInteger h) + 1/(contfrac t);

convergents :: [Integer] -> [Rational] ; convergents = (map contfrac) . inits