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

[Albert Chan]

(10-26-2021 02:13 PM)Ren Wrote:  What does ζ mean?

(10-25-2021 01:29 PM)Albert Chan Wrote:  ζ(2) = 1.65 - sum(1/(k^2*(4*k^2-1)), k=3..inf)

With denominator of O(k^4), this converge much faster than ζ(2) "definition"
How can we accelerate this with continued fraction ?

Only 1 term CF correction. It does help.

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