Infinite Expressions for Pi

• John Wallis (1655) took what can now be expressed as

  and without using the binomial theorem or integration (not invented yet) painstakingly came up with a formula for  to be

   .

• William Brouncker (ca. 1660's) rewrote Wallis' formula as a continued fraction, which Wallis and later Euler (1775) proved to be equivalent. It is unknown how Brouncker himself came up with the continued fraction,
   .

• James Gregory (1671) & Gottfried Leibniz (1674) used the series expansion of the arctangent function,
   ,

  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!

• History books credit Sir Isaac Newton (ca. 1730's) with using the series expansion of the arcsine function,
   ,

  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 .

• Leonard Euler (1748) proved the following equivalent relations for the square of ,
  
  

• Ko Hayashi (1989) found another infinite expression for  in terms of the Fibonacci numbers,
   .

原文链接 http://www.geom.uiuc.edu/~huberty/math5337/groupe/expresspi.html

PI=atan(1.0)*4;