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

[Albert Chan]

(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