This page lists a number of infinite expressions of . Proofs are not provided here, but the viewer is encouraged to study the sources listed in the reference page.
and without using the binomial theorem or integration (not invented yet) painstakingly came up with a formula for to be
and the fact that arctan(1) = /4 to obtain the series
Unfortunately, this series converges to slowly to be useful, as it takes over 300 terms to obtain a 2 decimal place precision. To obtain 100 decimal places of , one would need to use at least 10^50 terms of this expansion!
and the fact that arctan(1/2) = /6 to obtain the series
This arcsine series converges much faster than using the arctangent. (Actually, Newton used a slightly different expansion in his original text.) This expansion only needed 22 terms to obtain 16 decimal places for .
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http://www.geom.umn.edu/~huberty/math5337/groupe/expresspi.html Copyright © 1996-1997 Michael D. Huberty, Ko Hayashi & Chia Vang
Created: March 1996 ---- Last Modified: July 6, 1997
Copyright © 1996-1997 Michael D. Huberty, Ko Hayashi & Chia Vang