Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B]

[Gerson W. Barbosa]

(11-01-2021 12:56 AM)Albert Chan Wrote:  We can get the CF correction formula, using Euler–Maclaurin formula (see #9)

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Convert simple CF to generalized CF, we have (note: now N = n+0.5)

corr = 1/(N+ 1/(12N + 16/(5N + 81/(28*N + 256/(9*N + 5^4/(44*N + 6^4/(13*N + ...

Thank you very much! I am not an expert and cannot fully grasp what you have done, but it looks like you have a proof.

Almost three hundred years ago, before tackling the Basel problem, Euler evaluated the series to a few decimal digits using the Euler-MacLaurin method, incidentally the same method used in these RPL programs. The continued fraction correction appears to require less computational effort to obtain the same number of digits.