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

10262021, 08:28 PM
(This post was last modified: 10272021 01:21 PM by Albert Chan.)
Post: #9




RE: Evaluation of ζ(2) by the definition (sort of) [HP42S & HP71B]
(10262021 09:47 AM)Albert Chan Wrote: Interesting many CF correction formula involve the tag along +1/2 I get it now ! It is because correction is bounded by integral test. ∫(1/x^2, x=n .. inf) > Σ(1/k^2, k=n+1 .. inf) > ∫(1/x^2, x=n+1 .. inf) Σ(1/k^2, k=n+1 .. inf) ≈ ∫(1/x^2, x=n+0.5 .. inf) = 1/(n+0.5) Example, 2 terms + correction: ζ(2) ≈ 1 + 1/4 + 1/2.5 = 1.65 This explained why CF formula have form 1/((n+0.5) + ...) (10262021 03:24 PM)Albert Chan Wrote: CAS> zeta2(n) := 1.65  sum(1/(k^2*(4*k^21)), k=3..n) Above example, f(x) = 1/(x^2*(4*x^21) = 1/x^2  1/(x+1/2) + 1/(x1/2) ∫(f(x), x) = 1/x  (ln(x+1/2)  ln(x1/2)) = 1/x  2*atanh(1/(2x)) = 1/x  2*(1/(2x) + 1/(2x)^3/3 + ...) ≈ 1/(12*x^3) This explained corr2(n) denominator big term: 12*(n+0.5)^3 

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