Gregory's Series

Date: 06/14/97 at 01:21:49
From: Anonymous
Subject: Lost formula of the ages....

Dear Dr. Math,

College grad by 12 years seeks formula for pi.  Remember fragments as
1/3 + 1/5 - 1/7 + 1/9 ....  What is the whole sequence?

Thanks.

Date: 06/14/97 at 16:38:01
From: Doctor Anthony
Subject: Re: Lost formula of the ages....

We obtain Gregory's series from:  

  tan^(-1)(x) = INT(from 0 to x)[dx/(1+x^2)]

If we expand by the binomial theorem:  

  (1+x^2)^(-1) = 1 - x^2 + x^4 - x^6 + ...    

Integrating term by term:

  tan^(-1)(x) = x - x^3/3 + x^5/5 - x^7/7 + ....    

Now put in x = 1.  Then 

  tan^(-1)(1) = pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...

So pi =  4[1 - 1/3 + 1/5 - 1/7 + ....  to infinity].

Incidentally, this series converges VERY slowly and it is not a good 
series for finding pi to any degree of accuracy.  There are much 
better series available.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/