Question

can someone provide how i code this with matlab and provide explanations for each of them?

The natural logarithm of 1+x can be expressed using the Mercator series, given on the right-hand side of the equation below: Problem 2. (a) Generate a real random number in the range from-1.0 to 1.0 and assign it to variable x. (b) Compute the sum of the first 100 terms of the Mercator series fodthe r value obtained in part (a).

Answer 1

The problem is solved using MATLAB. The random number is generated using 'rand' function and a for loop is used to compute the sum of the first 100 terms. If you see the right hand side terms, the sign of each term alternates from plus to minus. If the power of x is odd, then the sign is plus and if the power of x is even, then the sign is minus.

The MATLAB code is as follows. Each command has comments

-------------- START OF MATLAB CODE -------------

%=========================================
% Mercator series
%=========================================
% The series is natural logrithm
% ln(1+x) = x - x^2/2 + x^3/3 - x^4 + .....
% We need to generate the sum for the first
% 100 terms

% (a) Generation of a random number between (-1.0 to 1.0)
% The rand function generates a real random value between (0.0 to 1.0)
% We need to make adjustments to match our requirements
% x = xmin + xrange*rand(1,1)
xmin = -1; % Minimum value in our requirement
xrange = 2; % Range of x in our requirement
x = xmin + xrange*rand(1,1); % A real random number is generated

% (b) Computation of mercator series
% Let y be the variable used for computation and representation of
% natural logarithm of (1+x)
% Number of terms is 100
N = 100; % Number of terms
y = 0; % Initial value
% Place a for loop for computation
for i=1:N
   % The power of x is odd put addition sign else subtraction sign
   if mod(i,2) ~= 0
       y = y + x^i/i;
   else
       y = y - x^i/i;
   end
end

% Display the generated random number
disp(x)
% Display the result of mercator series
disp(y)

%=========================================
% Verification of Mercator series
%=========================================
% Displaying the natural logarithm of x
disp(log(1+x))

------------- END OF MATLAB CODE --------------------

Save the above MATLAB code as 'mercator.m' file and run the file in the command window of MATLAB as 'mercator'. The results is displayed in the command window. The results which are displayed are

the first line is 'x', the second line is 'sum of terms' and the third line is 'natural logarithm'

Sample results are shown below after running the file in MATLAB command window,

0.66284
0.50852
0.50852

---------------------

0.58989
0.46366
0.46366

---------------

-0.85914
-1.9600
-1.9600

--------------

-0.45270
-0.60276
-0.60276

====================================================