Wallis' product exploration
02-08-2020, 07:13 PM
Post: #4
 Allen Member Posts: 227 Joined: Aug 2014
RE: Wallis' product exploration
I think the "slowness" of the clever asymptotic behavior of $$\prod_{k=1} ^{k=n} {4k^2 \over 4k^2-1}$$ can best be seen by factoring the numerator and denominator for the first few $$n$$ to see what is going on:

n=2
$$2^{6}\over 3^{2} 5^{1}$$

n=3
$$2^{8}\over 5^{2} 7^{1}$$

n=5
$$2^{16}\over 3^{4} 7^{2} 11^{1}$$

n=7
$$2^{22}\over 3^{3} 5^{1} 11^{2} 13^{2}$$

n=11
$$2^{38}\over 3^{2} 7^{2} 13^{2} 17^{2} 19^{2} 23^{1}$$

n=13
$$2^{46}\over 3^{3} 5^{4} 7^{2} 17^{2} 19^{2} 23^{2}$$

n=17
$$2^{64}\over 3^{6} 5^{3} 7^{1} 11^{2} 19^{2} 23^{2} 29^{2} 31^{2}$$

n=19
$$2^{70}\over 3^{3} 5^{4} 7^{2} 11^{2} 13^{1} 23^{2} 29^{2} 31^{2} 37^{2}$$

n=23
$$2^{84}\over 3^{6} 5^{4} 13^{2} 29^{2} 31^{2} 37^{2} 41^{2} 43^{2} 47^{1}$$

n=29
$$2^{108}\over 3^{2} 5^{2} 7^{2} 11^{2} 17^{2} 19^{2} 31^{2} 37^{2} 41^{2} 43^{2} 47^{2} 53^{2} 59^{1}$$

Do you notice anything interesting about the relationship between $$n$$ and the denominator when $$n$$ is prime?

What about when it's not prime?

n=28
$$2^{106}\over 3^{1} 5^{2} 7^{2} 11^{2} 17^{2} 19^{1} 29^{2} 31^{2} 37^{2} 41^{2} 43^{2} 47^{2} 53^{2}$$

n=29
$$2^{108}\over 3^{2} 5^{2} 7^{2} 11^{2} 17^{2} 19^{2} 31^{2} 37^{2} 41^{2} 43^{2} 47^{2} 53^{2} 59^{1}$$

n=30
$$2^{112}\over 7^{2} 11^{2} 17^{2} 19^{2} 31^{2} 37^{2} 41^{2} 43^{2} 47^{2} 53^{2} 59^{2} 61^{1}$$

n=31
$$2^{114}\over 3^{2} 7^{3} 11^{2} 17^{2} 19^{2} 37^{2} 41^{2} 43^{2} 47^{2} 53^{2} 59^{2} 61^{2}$$

n=32
$$2^{126}\over 3^{4} 5^{1} 7^{4} 11^{2} 13^{1} 17^{2} 19^{2} 37^{2} 41^{2} 43^{2} 47^{2} 53^{2} 59^{2} 61^{2}$$

n=33
$$2^{128}\over 3^{2} 5^{2} 7^{4} 13^{2} 17^{2} 19^{2} 37^{2} 41^{2} 43^{2} 47^{2} 53^{2} 59^{2} 61^{2} 67^{1}$$

n=34
$$2^{132}\over 3^{3} 5^{2} 7^{4} 13^{2} 19^{2} 23^{1} 37^{2} 41^{2} 43^{2} 47^{2} 53^{2} 59^{2} 61^{2} 67^{2}$$

Reminds me the highly composite binomial coefficients $$\binom{2n}{n}$$.

To represent as $$\pi$$ rather than $$\pi \over 2$$, one can just add 1 to the numerator like so:

$$\pi_{29} \approx \frac {2^{108+1}} {3^{2} 5^{2} 7^{2} 11^{2} 17^{2} 19^{2} 31^{2} 37^{2} 41^{2} 43^{2} 47^{2} 53^{2} 59^{1} }$$

17bii | 32s | 32sii | 41c | 41cv | 41cx | 42s | 48g | 48g+ | 48gx | 50g | 30b

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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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