Evaluation of ζ(2) by the definition (sort of) [HP42S & HP71B]

11032021, 01:14 AM
(This post was last modified: 11032021 07:13 AM by Albert Chan.)
Post: #19




RE: Evaluation of ζ(2) by the definition (sort of) [HP42S & HP71B]
(11032021 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 → 9.77190839357e09 XCas> e2 := dfc2f(makelist(1, 1, 21))  phi → 3.73253694619e09 XCas> abs(e1/e2) → 2.61803393629 

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