In 1671, the Scots mathematician James Gregory discovered the equivalent of the
ID: 3079762 • Letter: I
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 + 1/1.3 - 1/3.5 + 1/5.7 - 1/7.9 + ... By the way, one interesting fact is that yon 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) ... putting x = 1 ,, LHS = arctan(1) = pi/4 now putting x= 1 in RHS of the gregory series result follows...... b) .......now for the second using the given identity ...1/(1*3) = (1/2)*( (1-(1/3)) ......1/(3*5)===(1/2)*( (1/3)-(1/5))...... .1/(5*7)===(1/2)*( (1/5)-(1/7))......1/(7*9)===(1/2)*( (1/7)-(1/9)) and putting these terms in rhs of the equation we get (1/2)*[1+ ( (1-(1/3)) - ( (1/3)-(1/5)) +( (1/5)-(1/7)) -( (1/7)-(1/9)) +........ ] = (1/2)*[ 1+1 - (1/3) -(1/3) +(1/5) +(1/5) - (1/7) -(1/7) +.......] ===> [ 1 - (1/3) +(1/5) -(1/7)....... ] = pi/4_ ( from the first part of the question ) ...... hence proved hope this helps ............