[VA] SRC #010 - Pi Day 2022 Special

[Albert Chan]

(03-14-2022 08:03 PM)Valentin Albillo Wrote:  the awesome expression:

    

which beautifully relates \(\pi\) and e.

Below confirmed expression numerically, by turning sum to integral.

ln(pi)
= 3/2 + sum(1 + n^2 * ln(1 - 1/n^2), n = 2 .. inf)
= 3/2 + sum(1 - n^2 * ((1/n)^2 + (1/n)^4/2 + (1/n)^6/3 + (1/n)^8/4 + ...), n=2 .. inf)
= (1-ζ(0)) + (1-ζ(2))/2 + (1-ζ(4))/3 + (1-ζ(6))/4 + ...             // note: ζ(0) = -1/2

Zeta even integer generating function:



Replacing x by √x, and integrate from 0 to 1, we matched zeta terms.
We also need to add a function, to match fraction terms.

1/(1-x) = 1 + x + x^2 + ...
∫(1/(1-x), x=0..1) = 1 + 1/2 + 1/3 + ...

\(\displaystyle \ln(\pi) =
\int_0^1 \left(\frac{1}{1-x} + \frac{\pi\sqrt{x}}{2\;\tan(\pi\sqrt{x})}\right) dx\)

Note that integrand is inaccurate when x approach 1. P cannot be set too small.

10 P=.000001
20 DEF FNF(X,Y)=1/(1-X)+.5*Y/TAN(Y)
30 DISP INTEGRAL(0,1,P,FNF(IVAR,PI*SQRT(IVAR))), EXP(RES)
>
>RUN
1.14472988295       3.14159264448