Gregory's SeriesDate: 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/ |
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