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

11012021, 12:56 AM
Post: #14




RE: Evaluation of ζ(2) by the definition (sort of) [HP42S & HP71B]
We can get the CF correction formula, using Euler–Maclaurin formula (see #9)
XCas> C(k) := bernoulli(k)/k!; // EulerMaclaurin 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 N1/2 XCas> [top,bot] := f2nd(1/corr(N=N+1/2)) :; XCas> q:=quo(top,bot,N); [top,bot] := [bot,topq*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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