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Infinite Product for pi/2 by Wallis

This infinite product for pi/2 was first published by the English mathematiciian John Wallis in 1655. Dr. Wallis made significant contributions to mathematics. Among which he is credited with introducing our modern symbol for infinity.

John Wallis(1616-1703) http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Wallis.html

Wiki for "Wallis Product" https://en.wikipedia.org/wiki/Wallis_product

"An Elementary Proof of Wallis' Product Forula for Pi" https://fermatslibrary.com/s/... ...an-elementary-proof-of-wallis-product-formula-for-pi

First 8 terms, expanded form:

piover2_exp 22^42^* 62^* 82^* 102^* 122^* 142^* 12^32^* 52^* 72^* 92^* 112^* 132^* 15 * / :

piover2_exp 1.51933681444171

π 2 / 1.5707963267949

piover2_exp 2 * 3.03867362888342

π 3.14159265358979

This infinite product converges to pi very slowly. It is not a practical way to calculate pi, but it is quite interesting.

In Product Notation, the Wallis pi/2 formula is:

piover2_prod 4 n 2^* 4 n 2^* 1 - / (n 110000 product :

piover2_prod 1.57075705934097

piover2_prod 2 * 3.14151411868194

π 3.14159265358979

Examining the convergence to the value of pi as the number of terms gets very large below. It is very, very slow to converge. The relatively fast PC used for this calcuation below took about 40 seconds to complete. One estimate I found on the web was that it take 100 million terms to reach 8 digit precision. I believe it!

terms 1000100005000010000020000050000061 mat :

piover2_converge 4 n 2^* 4 n 2^* 1 - / (n 11000 product 4 n 2^* 4 n 2^* 1 - / (n 110000 product 4 n 2^* 4 n 2^* 1 - / (n 150000 product 4 n 2^* 4 n 2^* 1 - / (n 1100000 product 4 n 2^* 4 n 2^* 1 - / (n 1200000 product 4 n 2^* 4 n 2^* 1 - / (n 1500000 product 61 mat :

piover2_converge 1.57040387301519 1.57075705934097 1.57078847291147 1.57079239982868 1.57079436330568 1.57079554139774 61 mat

π 2 / 1.5707963267949