e^x maclaurin serie

Hello everyone, I'm very new to MATLAB. Could you help me with this question please?

How to write codes that calculate e^x by serializing the x value entered from the keyboard into a series equal to the number of terms (N) entered from the keyboard?

[Dyuman Joshi]

To clarify, you want to find the Maclauren series of exponential function upto N terms? If yes, do you want to do that numerically or symbolically?

[Sam Chak]

Hi @Firuze, I don't remember the Maclaurin series for . Could you help me with typing out the Maclaurin series please?

Click this icon and enter the series formula:

%% Maclaurin series for e^x
expx = @(x) ...

[Sam Chak]

Hi @Firuze,

I realized that I can use the taylor() function to compute the Maclaurin series without having to memorize the series expansion. There are several solutions, depending on whether certain built-in functions can be used or not in your homework.

%% Get input values from the Keyboard User
% x = input('Enter the value of x: ');
% N = input('Enter the number of terms (N): ');
x = 1;
N = 5;

sympref('PolynomialDisplayStyle', 'ascend');
syms x

%% Display the series expansion for e^x
T     = taylor(exp(x), x, 'Order', N+1)     % Unsure how you define the number of terms

T = 

%% Evaluate the series
Teval = double(subs(T, x, 1))

Teval = 2.7167

[Firuze]

thank you

Answer 1

[Hassaan]

@Firuze

Numerically:

% Prompt the user to enter the value of x
x = input('Enter the value of x: ');

% Prompt the user to enter the number of terms N
N = input('Enter the number of terms N: ');

% Initialize the result of e^x
ex = 1; % The first term of the series is always 1

% Calculate e^x using the series expansion [for-loop]
for n = 1:N
    term = x^n / factorial(n); % Calculate each term in the series
    ex = ex + term; % Add each term to the result
end

% % Calculate e^x using the series expansion [vectorization]
% n = 0:N; 
% ex = sum(x.^n ./ factorial(n));

% Display the result
fprintf('e^x calculated using the series expansion is: %f\n', ex);

% For comparison, calculate e^x using MATLAB's exp function
ex_true = exp(x);
fprintf('e^x calculated using MATLAB''s exp function is: %f\n', ex_true);

It will prompt you to input the value of x and the number of terms N. Then, it will calculate and display the result of e^x using the series expansion up to N terms. Additionally, it shows the result calculated by MATLAB's built-in exp function for comparison.

Output

Enter the value of x: 1

Enter the number of terms N: 5

e^x calculated using the series expansion is: 2.716667

e^x calculated using MATLAB's exp function is: 2.718282

Symbol Evaluation

% Ask for input from the keyboard
x = input('Enter the value of x: ');
N = input('Enter the number of terms N: ');

% Initialize the symbolic variable and the result
syms n;
ex = sym(0);  % Symbolic 0

% Calculate the series expansion symbolically
for n = 0:N-1
    ex = ex + (x^n)/factorial(n);
end

% Display the symbolic result
disp(ex);

Symbolic Expression

% Define the symbolic variable x and the summation index n
syms x n;

% Ask for input for the number of terms in the series N
N = input('Enter the number of terms N: ');

% Define the symbolic expression for the series expansion of e^x
e_x_series = symsum(x^n/factorial(n), n, 0, N-1);

disp(e_x_series)

% Pretty print the symbolic series expansion
% pretty(e_x_series)

Output

Enter the number of terms N: 5

x^4/24 + x^3/6 + x^2/2 + x + 1

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[the cyclist]

This can be vectorized by replacing

% Initialize the result of e^x
ex = 1; % The first term of the series is always 1

% Calculate e^x using the series expansion
for n = 1:N
    term = x^n / factorial(n); % Calculate each term in the series
    ex = ex + term; % Add each term to the result
end

with

n = 0:N;
ex = sum(x.^n ./ factorial(n));

[Hassaan]

@the cyclist Agreed another way to achieve same results.

[John D'Errico]

Please do not do obvious homework assignments for students who show no effort. This does not help the student. It teaches them only that there is always some sucker willing to do their homework for them.

Worse, it teaches that same student to then keep on posting additional homework assignments, thinking they have landed on a homework gold mine.

And finally, it teaches other students to follow their lead. This damages the forum itslef, as we will be overwhelmed with students posting their homework.

[Hassaan]

@John D'Errico I do agree and I try to avoid as much as possible such scenarios.

Answer 2

[Sulaymon Eshkabilov]

A function can be written to compute Mclaurin series approximation, e.g.:

% E.g.: approximation of exp(x) at x = pi
x = pi; 
N_terms = 10; % Number of terms in the series

Solution = Maclaurin(x, N_terms);

fprintf('Maclaurin series approx. for exp(%g) with %d terms: %g\n', x, N_terms, Solution);

function function SOL = Maclaurin(x, terms)
    n = 0:terms;
    SOL = sum((x.^n) ./ factorial(n));
end

Alt. Solution (computationally slow)

% E.g.: approximation of exp(x) at x = pi
x = pi; 
N_terms = 10; % Number of terms in the series

Solution = Maclaurin2(x, N_terms);

fprintf('Maclaurin series approx. for exp(%g) with %d terms: %g\n', x, N_terms, Solution);
function SOL = Maclaurin2(x, terms)
     
    SOL = 1; % Initialize with the first term
    
    for n = 1:terms
        SOL = SOL + x^n / factorial(n);
    end
end

[John D'Errico]

Please do not do obvious homework assignments for students who show no effort. This does not help the student. It teaches them only that there is always some sucker willing to do their homework for them.

Worse, it teaches that same student to then keep on posting additional homework assignments, thinking they have landed on a homework gold mine.

And finally, it teaches other students to follow their lead. This damages the forum itslef, as we will be overwhelmed with students posting their homework.