Wallis' product exploration

[Gerson W. Barbosa]

Try the following for even n and see what you get.


\(\frac{\pi }{2}\approx \left ( \frac{4}{3} \cdot \frac{16}{15}\cdot \frac{36}{35}\cdot\frac{64}{63} \cdots \frac{ 4n ^{2}}{ 4n ^{2}-1}\right )\left ( 1+\frac{1}{4n+\frac{3}{2-\frac{1}{4n+\frac{5}{2-\frac{3}{4n+\frac{7}{2-\frac{5}{4n+\frac{9}{2-\frac{7}{\dots \frac{ \ddots }{2-\frac{n-3}{4n+\frac{n+1}{2}}}}}}}}}}}} \right )\)



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P.S.:

This will handle both parities:

\(\frac{\pi }{2}\approx \left ( \frac{4}{3} \times \frac{16}{15}\times \frac{36}{35}\times\frac{64}{63} \times \cdots \times \frac{ 4n ^{2}}{ 4n ^{2}-1}\right )\left ( 1+\frac{1}{4n+\frac{3}{2-\frac{1}{4n+\frac{5}{2-\frac{3}{4n+\frac{7}{2-\frac{5}{4n+\frac{9}{2-\frac{7}{\dots \frac{ \ddots }{4n \left ( n \bmod 2\right) + 2\left ( n+1 \bmod 2 \right )+\frac{1-n+\left ( 2n+1 \right )\left ( n \bmod 2 \right )}{4n \left (n+1 \bmod 2\right)
+ 2\left ( n \bmod 2 \right )}}}}}}}}}}} \right )\)

This approximation gives \(\frac{4}{3}n\) correct significant digits.