Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B]
10-26-2021, 08:28 PM (This post was last modified: 10-27-2021 01:21 PM by Albert Chan.)
Post: #9
 Albert Chan Senior Member Posts: 2,516 Joined: Jul 2018
RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B]
(10-26-2021 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) + ...)

(10-26-2021 03:24 PM)Albert Chan Wrote:  CAS> zeta2(n) := 1.65 - sum(1/(k^2*(4*k^2-1)), k=3..n)
CAS> corr2(n) := -1/horner([12,0,4.2,0],n+0.5)

CAS> zeta2(10) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 1.6450058264
CAS> Ans+corr2(10) → 1.64493406774
CAS> pi*pi/6. ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 1.64493406685

Above example, f(x) = 1/(x^2*(4*x^2-1) = -1/x^2 - 1/(x+1/2) + 1/(x-1/2)

∫(f(x), x)
= 1/x - (ln(x+1/2) - ln(x-1/2))
= 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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 Messages In This Thread Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Gerson W. Barbosa - 10-23-2021, 02:49 PM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Albert Chan - 10-25-2021, 01:29 PM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Gerson W. Barbosa - 10-26-2021, 02:12 AM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Albert Chan - 10-26-2021, 09:47 AM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Albert Chan - 10-26-2021 08:28 PM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Albert Chan - 10-29-2021, 02:16 PM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Albert Chan - 11-01-2021, 10:42 PM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Gerson W. Barbosa - 11-02-2021, 12:28 PM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Albert Chan - 10-27-2021, 05:12 PM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Albert Chan - 11-04-2021, 08:35 PM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Ren - 10-26-2021, 02:13 PM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - floppy - 10-26-2021, 03:04 PM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Albert Chan - 10-26-2021, 03:24 PM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Gerson W. Barbosa - 10-26-2021, 03:58 PM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Ren - 10-27-2021, 01:32 PM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Albert Chan - 10-31-2021, 03:40 PM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Albert Chan - 11-05-2021, 03:55 PM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Albert Chan - 11-01-2021, 12:56 AM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Gerson W. Barbosa - 11-01-2021, 05:04 AM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Albert Chan - 11-03-2021, 12:38 AM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Albert Chan - 11-03-2021, 01:14 AM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Albert Chan - 11-03-2021, 11:28 PM RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B] - Albert Chan - 11-04-2021, 10:42 PM

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