Wallis' product exploration

[Albert Chan]

You can check the result with a direct formula, for n terms, here

Or, using lgamma(), we have

\(\log \left(2 \prod _{k=1} ^{k=n} {4k^2 \over 4k^2-1}\right) =
4\log(\Gamma(n)) - 2\log(\Gamma(2n)) + n \log(16) + 2 \log(n) - \log(4n+2) \)

Example, using log1p and sum the logarithm of the terms backwards, this matched above formula

lua> require'mathx'
lua> log1p = mathx.log1p
lua> s=0; for i=100e6, 1, -1 do s = s - log1p(-1/(4*i*i)) end
lua> = 2*math.exp(s)
3.1415926457358117