In 1671, the Scots mathematician James Gregory discovered the equivalent of the

Question

In 1671, the Scots mathematician James Gregory discovered the equivalent of the inverse tangent series: expressed in modern terms, tan-1 x = x - x3/3 + x5/5 - x7/7 + x9/9 - ..., |x| 1. Use this series to obtain Leibniz's famous alternating series for pi/4, a fact apparently overlooked by Gregory: pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ... By using a formula 1/(2n + 1)(2n + 3) = 1/2 (1/2n + 1 - 1/2n + 3), prove another formula for pi pi/4 = 1/2 + 1/1 middot 3 - 1/ 3 middot 5 + 1/5 middot 7 - 1/7 middot 9 + ... By the way, one interesting fact is that you can 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), but the series in (a) is so slowly converging such that one has to calculate several hundred terms in reach the same accuracy.

Explanation / Answer

a)
sub x=1 in that formula we get
atan(1)=1-1/3+1/5-1/7+...........
=>/4=1-1/3+1/5-1/7+...........

hence showed as atan(1)=/4=>tan/4 =1

b)given

1/(2n+1)(2n+3) = .5(1/(2n+1) -1/(2n+3))

=>/4= 1-1/3+1/5-1/7+...........

=.5+.5-.5 1/3-.5*1/3+.5*1/5+.5*1/5-.5*1/7-.5*1/7+........................

=.5+.5(1/1 -1/3)-.5(1/3 -1/5)+.5(1/5 -1/7)+...........

=.5+1/1*3 -1/3*5+1/(5*7)+..............

hence showed

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