Yet another π formula

[Gerson W. Barbosa]

(01-07-2021 09:56 PM)Albert Chan Wrote:  

(01-07-2021 12:57 PM)Gerson W. Barbosa Wrote:  Great! The continued fraction part is left unproved, however. But that would not be easy, I presume.

"Wasicki's formula" had first term converge to pi, correction converge to 0. QED

Were the correction term simply 1/(2n*(n + 1)) then this proof would still hold. Perhaps I should say “prove that the continuous fraction is equivalent to the optimal correction polynomial expression to the series”. That would be really hard, at least for me. If you find a valid proof then the formula should be renamed as “Wasicki-Chan Formula” or more appropriately “Chan-Wasicki Formula”. Anyone can easily find such formulae. Anyone or anything – even Mathematica and W/A can, I think. But only a few can prove them. Anyway nowadays these formulae are just useless mathematical curiosities.

(01-07-2021 09:56 PM)Albert Chan Wrote:  As to the gain of 25/12 digits per term, it tested OK even with 10,000 digits precision.

Thank you for performing these tests. Much appreciated!