Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B]

[Albert Chan]

(10-25-2021 01:29 PM)Albert Chan Wrote:  Convergence speed very impressive, n=8 converged to float(pi^2/6)

2nd version, by putting some reasonable number to b, before loop start (*)
Now, n=6 already converged to float(pi^2/6)

Code:

function zeta2(n)
    local N, c, t = n+0.5, (n%2)*6+2
    local N2 = N*N
    local a, b = 0, (N2+N) * 0.93/(10-c)
    for i = n, 2, -1 do
        t = i*i
        a = a + 1/t
        b = t*t/(b + c*N2)
        c, N2 = 10-c, N2-N
    end
    return 1/(N+1/(b+c*N2)) + a + 1
end

lua> for n=0,8 do print(n, zeta2(n)) end
0      1.4539195555018722
1      1.6448912932530366
2      1.6449340922532778
3      1.6449340670471453
4      1.6449340668467034
5      1.6449340668482317
6      1.6449340668482264
7      1.6449340668482264
8      1.6449340668482264

(*) this is how initial b is estimated, by looking ahead.

XCAS> b2 := (N+3/2)^4 / (c*N*(N+2))
XCAS> b1 := (N+1/2)^4 / ((10-c)*N*(N+1) + b2)

XCAS> simplify(partfrac(b1(c=8))[1])       → (2312*N^2+1904*N+480)/4913
XCAS> simplify(partfrac(b1(c=2))[1])       → (578*N^2+476*N+120)/4913

2312/4913*2 ≈ 0.9412
If N is large, this should be the constant to use instead of 0.93

Update: Looking ahead a few more, 0.944 is about optimum