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Many series are used to calculate the value of pi.  In this blog, we compare two series, one by Gregory and another by Ramanujan.

Here is a MATLAB program that does the comparison for you.  The MATLAB program can be downloaded as a Mfile (better to download it, as single quotes from the web-post do not translate correctly with the MATLAB editor).  The html file showing the mfile and the command window output is also available.

%% COMPARING TWO SERIES FOR VALUE OF PI
% Language : Matlab 2007a
% Authors : Autar Kaw
% Last Revised : October 30, 2008
% Abstract: This program compares results for the value of
% pi using a) Gregory series and b) Ramanajun series
clc
clear all
clf
format long
disp(‘This program compares results for the value of’)
disp(‘pi using a) Gregory series and b) Ramanajun series’)
disp(‘  ‘)
disp(‘Gregory series’)
disp(‘pi=sum over k from 0 to inf of (4*((-1)^k/(2*k+1))’)
disp(‘  ‘)
disp(‘Ramanajun Series’)
disp(‘1/pi=sum over k from 0 to infinity of 2*sqrt(2)/9801*((4k)!*(1103+26390k)/(k!)^4*396^(4*k))’)

%% INPUTS.
%If you want to experiment this the only parameter
% you should and can change.
% Maximum number of terms
n=30;

%% PROGRAM

%% GREGORY SERIES
pi_gregory=0;
for i=1:1:n
pi_gregory=pi_gregory+(-1)^(i+1)*4*(1/(2*i-1));
pi_gregory_array(i)=pi_gregory;
end

%% RAMANUJAN SERIES
pi_ram=0;
for i=0:1:n-1
pi_ram=pi_ram+2*sqrt(2)/9801.0*(factorial(4*i))*(1103.0+26390.0*i)/((factorial(i)^4)*(396)^(4*i));
pi_ram_array(i+1)=1/pi_ram;
end

%% THE OUTPUT
disp(‘ ‘)
fprintf(‘\nNumber of Terms = %g’,n)
fprintf(‘\nGregory Series Value = %g’,pi_gregory)
fprintf(‘\nRamanujan Series Value = %g’,1/pi_ram)
disp( ‘   ‘)

%% PLOTTING THE TWO SERIES AS A FUNCTION OF TERMS
x=1:1:n;
hold on
xlabel(‘Number of terms’)
ylabel(‘Value of pi’)
title(‘Comparing Gregory and Ramanujan series’)
plot(x,pi_gregory_array,’color’,’blue’,’LineWidth’,2)
hold on
plot(x,pi_ram_array,’color’,’black’,’LineWidth’,2)
legend(‘Gregory Series’,’Ramanajun Series’,1)

This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://numericalmethods.eng.usf.edu.

An abridged (for low cost) book on Numerical Methods with Applications will be in print (includes problem sets, TOC, index) on December 10, 2008 and available at lulu storefront.

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