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

[Gerson W. Barbosa]

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.

HP-42S/Free42 program:

Code:


00 { 65-Byte Prgm }
01▸LBL "z"
02 0.5
03 +
04 STO 01
05 IP
06 0
07 STO 02
08▸LBL 00
09 RCL ST Y
10 X↑2
11 1/X
12 +
13 -1
14 RCL ST Z
15 Y↑X
16 +/-
17 3
18 ×
19 5
20 +
21 0.5
22 RCL+ ST T
23 ×
24 RCL× 01
25 RCL+ 02
26 R↑
27 X↑2
28 X↑2
29 X<>Y
30 ÷
31 STO 02
32 R↓
33 DSE ST Y
34 GTO 00
35 RCL 02
36 RCL+ 01
37 1/X
38 +
39 END

n = 12 on the HP-42S and n = 16 on Free42 will suffice for 12 and 34 correct digits, respectively.

6 XEQ “z” → 1.64493406685

16 XEQ “z” →

1.644934066848226436472415166646025

HP-71B BASIC program;

Code:


10 S=0
15 C=0
20 INPUT N
25 K=N+.5
30 A=3-6*MOD(N,2)
35 FOR I=N TO 1 STEP -1
40 S=S+1/(I*I)
45 C=I^4/(C+K*(5-A)*(I+.5))
50 A=-A
55 NEXT I
60 DISP S+1/(K+C)

RUN

? 6

1.64493406685

Interested readers are invited to provide

- optimized versions of the given programs;

- versions for other calculator, such as the HP-41;

- 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.