In 1671, the Scots mathematician James Gregory discovered the equivalent of the
ID: 2964886 • Letter: I
Question
In 1671, the Scots mathematician James Gregory discovered the equivalent of the inverse tangent series: expressed in modern terms, Use this series to obtain Leibniz s famous alternating series for pi / 4, a fact apparently overlooked by Gregory: Show that pi / 4 may also be represented by the series Approximate the value of pi with an accuracy of 0.001 by calculating the sum of the first 10 terms of the series in (b); the slowly converging Leibniz series would require several hundred terms to achieve the same accuracy.Explanation / Answer
(a)
arctanx = x-x^3/3 +x^5/5-....
put x = 1
=>
arctan1 = 1-1^3/3 +1^5/5 -.... = 1-1/3+1/5-1/7....
=>
pi/4 = 1-1/3=1/5 -1/7.....
thus proved
(c)
pi = 4*[1/2 +1*1/3+1/3*1/5 +1/5*1/7+1/7*1/9 +1/9*1/11 +1/11*1/13 +1/13*1/15 +1/15*1/17+1/17*1/19] = 3.894
8.4)
pi/4 = 1-1/3+1/5-1/7....
= (1-1/3)+(1/5-1/7) +(1/9-1/11)+...
= (2)/1.3 +2/5.7+2/9.11+....
= 2[1/1.3 +1/5.7+1/9.11+..]
=>
pi/8 = 1/1.3+1/5.7+1/9.11
thus proved