Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B]
11-01-2021, 12:56 AM
Post: #14
 Albert Chan Senior Member Posts: 2,516 Joined: Jul 2018
RE: Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B]
We can get the CF correction formula, using Euler–Maclaurin formula (see #9)

XCas> C(k) := bernoulli(k)/k!; // Euler-Maclaurin formula coefs
XCas> f, a, b := 1/x^2, N, inf; // N = n+1

XCas> corr := int(f,x,a,b) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 1/N
XCas> corr += preval(f,a,b) * (-1/2) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 1/N+1/(N^2*2)
XCas> f := f' :; corr += preval(f,a,b) * C(k:=2)
XCas> f := f'':; corr += preval(f,a,b) * C(k+=2)

Run last line a few times, we have this corr, as polynomial of 1/N:

XCas> e2r(corr(N=1/x))

[-691/2730, 0, 5/66, 0, -1/30, 0, 1/42, 0, -1/30, 0, 1/6, 1/2, 1, 0]

Confirm numerically:

XCas> n:=5; sum(1./k^2, k=1..n) + corr(N=n+1), pi*pi/6.

(5, 1.64493406685, 1.64493406685)

Now, we are ready to convert corr into CF formula.
corr assumed N=n+1, but we wanted N=n+1/2, so we shift, and flip it.
Note, we replace N by N+1/2 in 1 step, instead of N by n+1 then n by N-1/2

XCas> [top,bot] := f2nd(1/corr(N=N+1/2)) :;
XCas> q:=quo(top,bot,N); [top,bot] := [bot,top-q*bot]:; → N

Run last line a few more times, we have the other quotients:

12*N, 5/16*N, 448/81*N, 729/4096*N, 180224/50625*N, 8125/65536*N

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 + ...
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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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