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

10232021, 02:49 PM
Post: #1




Evaluation of ζ(2) by the definition (sort of) [HP42S & HP71B]
FWIW,
A new method for the fast evaluation of ζ(2) using the definition as a basis The series, 1 + 1/4 + 1/9 + 1/25 + 1/36 + 1/49…, converges very slowly to the exact result, π^2/6. In order to obtain n correct digits the series should be evaluated up to the (10^n)th term. However, the addition of n+1 terms of a simple continued fraction after the evaluation of the first n terms of the series will significantly speed up the rate of convergence, yielding slightly more than 2n correct digits. For example, for n = 3, 1+1/4+1/9+1/((3+1/2)+1/(12*(3+1/2)+16/(5*(3+1/2)+81/(28*(3+1/2))))) = 55783/33912 = 1.6449339 The coefficients of the denominators of the continued fraction, 12, 5, 28, 9, 44, 13…, obey the formula k(i) = (5  3*(1)^i)*(i + 1/2). The numerators, 1, 16, 81, 256, 625, 1296…, are quite obvious. HP42S/Free42 program: Code:
n = 12 on the HP42S and n = 16 on Free42 will suffice for 12 and 34 correct digits, respectively. 6 XEQ “z” → 1.64493406685 16 XEQ “z” → 1.644934066848226436472415166646025 HP71B BASIC program; Code:
RUN ? 6 1.64493406685 Interested readers are invited to provide  optimized versions of the given programs;  versions for other calculator, such as the HP41;  a proof (I don’t have any – this is the result of a Friday afternoon work only, which until minutes ago I thought to be a Saturday afternoon. Still looking like Sunday morning to me). Pointing out typos and mistakes, either math or grammar related, are welcome. Gerson. 

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