Infinite Expressions for Pi

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

  And without using the binomial theorem or integration (not invented yet) painstakingly came up with a formula

  .

• William Brouncker (ca. 1660 ' s) rewrote Wallis ' formula as a continued fraction, which Wallis and later Euler (1775) proved To is equivalent. It's 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, arctan (1) = /4 to obtain the series

   .

  Unfortunately, this series converges to slowly to being useful, as it takes over terms to obtain a 2 decimal Place precision. To obtain decimal places of , one would need to use at least 10^50 terms of this expansion!

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

  And the fact that's arctan =/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-needed-terms to obtain-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,
  .

Original linkHttp://www.geom.uiuc.edu/~huberty/math5337/groupe/expresspi.html
Pi=atan (1.0);

Infinite Expressions for Pi