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

[Albert Chan]

(11-03-2021 12:38 AM)Albert Chan Wrote:  However, accuracy is less by 0.4180 digit, compared with non alternating sum.

Accuracy gap of 2/5 of a digit start from the beginning, from n=1.
For n=400, gap = 0.417975 digit, which is very close to log10(1+ϕ) ≈ 0.4179752805

I don't know if this is related, but it is the same ratio if we add 1 more CF term to estimate ϕ

ϕ = [1; 1, 1, 1, 1, ...] = [1;ϕ] = [1;1,ϕ] = [1;1,1,ϕ] = ...

Adding 1 CF term will make estimate better, roughly by denominator square, ϕ^2 = 1+ϕ

XCas> phi := (1+sqrt(5))/2.                           → 1.61803398875
XCas> e1 := dfc2f(makelist(1, 1, 20)) - phi      → 9.77190839357e-09
XCas> e2 := dfc2f(makelist(1, 1, 21)) - phi      → -3.73253694619e-09
XCas> abs(e1/e2)                                          → 2.61803393629