Yet another π formula

[Albert Chan]

(11-25-2024 01:47 PM)Albert Chan Wrote:  To match same number of 'terms', we set n = 5-1 = 4 , d = 6-1 = 5
Note: S=S1-S2 because n is even, otherwise it would be S=S1+S2

lua> S1 = 1/(1*3) - 1/(3*5) + 1/(5*7) - 1/(7*9) + 1/(9*11)
lua> S2 = 0.5/(11*13) * (1 + 2/15*(1 + 3/17*(1 + 4/19*(1 + 5/21*(1 + 6/23)))))
lua> (S1-S2)*4 + 2
3.1415943802536885

FYI, above convergence acceleration scheme is EXACTLY the same as my previous post.

pi/2 = (2 - 2/3 + 2/5 - 2/7 + 2/9) - 9!! * sum(k!/(2k+11)!!, k=0..inf)

lua> s1 = (2 - 2/3 + 2/5 - 2/7 + 2/9)
lua> s2 = 1/11*(1 + 1/13*(1 + 2/15*(1 + 3/17*(1 + 4/19*(1 + 5/21*(1 + 6/23))))))
lua> (s1-s2)*2
3.141594380253689