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

10262021, 09:47 AM
Post: #4




RE: Evaluation of ζ(2) by the definition (sort of) [HP42S & HP71B]
(10262021 02:12 AM)Gerson W. Barbosa Wrote: Numerically. I computed the difference of the exact result and the partial sums for a few n and examined their inverses. Interesting many CF correction formula involve the tag along +1/2 Quote:By doing the calculation with 34 digits, it was possible to get the next three terms: Impressive you can deduce denominator pattern with only 4 numbers: 1, 12, 5, 28, ... Anther way to check units (dimensional analysis). Numerator: 1, 16, 81, ... seems to involve unit of U^4 Denominator should have unit U^2, to have everything consistent. But we only have unit U from (n+1/2), so coefficients somehow also have unit of U To pull units off coefficients, we interpolate: (1,1), (3,5) ⇒ y = 2*x1 = 2*(x0.5) (2,12), (4,28) ⇒ y= 8*x4 = 8*(x0.5) It would be better if we had more coefficients to confirm the pattern. But, this suggested "unit free" coefficients are alternating 2, 8, 2, 8 ... 1/(2*0.5*(n+0.5) + 1/(8*1.5*(n+0.5) + 16/(2*2.5*(n+0.5) + 81/(8*3.5*(n+0.5) + ... )))) Quote:(10252021 01:29 PM)Albert Chan Wrote: This converge even faster, A cute sum of Ramanujan Considering number of terms needed to sum 1/k^2, this is a huge improvement. Maybe we can accelerate convergence with CF correction ? 

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