Wallis' product exploration
02-11-2020, 04:11 PM (This post was last modified: 06-27-2020 04:29 PM by Gerson W. Barbosa.)
Post: #17
 Gerson W. Barbosa Senior Member Posts: 1,604 Joined: Dec 2013
RE: Wallis' product exploration
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.
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 Messages In This Thread Wallis' product exploration - pinkman - 02-07-2020, 10:55 PM RE: Wallis' product exploration - Albert Chan - 02-08-2020, 02:13 AM RE: Wallis' product exploration - pinkman - 02-08-2020, 02:42 PM RE: Wallis' product exploration - Allen - 02-08-2020, 07:13 PM RE: Wallis' product exploration - Albert Chan - 02-08-2020, 07:53 PM RE: Wallis' product exploration - Allen - 02-08-2020, 08:58 PM RE: Wallis' product exploration - pinkman - 02-09-2020, 05:44 AM RE: Wallis' product exploration - Allen - 02-08-2020, 08:13 PM RE: Wallis' product exploration - Allen - 02-09-2020, 01:08 PM RE: Wallis' product exploration - Allen - 02-09-2020, 01:57 PM RE: Wallis' product exploration - Allen - 02-09-2020, 03:08 PM RE: Wallis' product exploration - pinkman - 02-09-2020, 02:14 PM RE: Wallis' product exploration - Gerson W. Barbosa - 02-09-2020, 10:58 PM RE: Wallis' product exploration - Gerson W. Barbosa - 02-11-2020, 12:53 AM RE: Wallis' product exploration - EdS2 - 02-10-2020, 10:35 AM RE: Wallis' product exploration - pinkman - 02-11-2020, 10:02 AM RE: Wallis' product exploration - Gerson W. Barbosa - 02-11-2020 04:11 PM RE: Wallis' product exploration - Gerson W. Barbosa - 02-11-2020, 10:48 PM RE: Wallis' product exploration - pinkman - 02-12-2020, 10:01 PM RE: Wallis' product exploration - Gerson W. Barbosa - 02-13-2020, 12:17 AM

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