Wallis' product exploration
02-13-2020, 12:17 AM
Post: #20
 Gerson W. Barbosa Senior Member Posts: 1,604 Joined: Dec 2013
RE: Wallis' product exploration
(02-12-2020 10:01 PM)pinkman Wrote:
(02-11-2020 10:48 PM)Gerson W. Barbosa Wrote:  Only linear convergence, but better than waiting forever for a just few digits.

Thank you for all, this is beautiful.

Thank you and your cousin for having started that conversation about the Wallis Product and posting here, otherwise I probably wouldnâ€™t have tried. Unlike approximation polynomials, the equivalent continued fractions usually follow beautiful patterns. Once you have a continued fraction, those polynomials are just their successive approximants. For instance,

Simplify [1 + 1/(4 n + 3/(2 - 1/(4 n + 5/2)))]

-> (64 n^2 + 72 n + 23)/(64 n^2 + 56 n + 15)

= (n^2 + 9n/8 + 23/64)/(n^2 + 7n/8 + 15/64)

This technique had worked previously for other slowly convergent series, so I guessed it would work for Wallis as well.
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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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