The value of Euler’s number e can be calculated with the series

The Correct Answer and Explanation is:

To calculate Euler’s number eee using the series expansion, we use the following formula:e=∑n=0∞1n!e = \sum_{n=0}^{\infty} \frac{1}{n!}e=n=0∑∞​n!1​

This series converges quickly, so we can approximate eee by summing the first NNN terms. The true value of eee is exp⁡(1.0)\exp(1.0)exp(1.0) in MATLAB, and we can compute the relative error as follows:True Relative Error=∣eapprox−etrue∣etrue\text{True Relative Error} = \frac{|e_{\text{approx}} – e_{\text{true}}|}{e_{\text{true}}}True Relative Error=etrue​∣eapprox​−etrue​∣​

Now, let’s write the MATLAB function that will compute the estimate of eee for a given NNN and the corresponding relative error.

MATLAB Code:

matlabCopyEditfunction [e_estimate, true_rel_error] = estimate_e(N)
    % Function to estimate Euler's number e using the first N terms
    % of the series and calculate the true relative error
    
    % Initialize the estimate
    e_estimate = 0;
    
    % Sum the first N terms of the series
    for n = 0:N-1
        e_estimate = e_estimate + 1/factorial(n);
    end
    
    % Calculate the true value of e using exp(1.0)
    e_true = exp(1.0);
    
    % Calculate the true relative error
    true_rel_error = abs(e_estimate - e_true) / e_true;
    
    % Display results
    disp(['Estimate of e for N = ', num2str(N), ' : ', num2str(e_estimate)]);
    disp(['True Relative Error: ', num2str(true_rel_error)]);
end

Explanation:

• The function estimate_e(N) takes an integer NNN, which specifies the number of terms to be used in the series.
• The loop sums the terms of the series 1n!\frac{1}{n!}n!1​ for n=0n = 0n=0 to N−1N-1N−1.
• The true value of eee is computed using exp(1.0) in MATLAB.
• The relative error is calculated using the formula for true relative error.

Example for different values of NNN:

You can run the function for different values of NNN as follows:

a) N=10N = 10N=10:

matlabCopyEdit[e_10, error_10] = estimate_e(10);

b) N=20N = 20N=20:

matlabCopyEdit[e_20, error_20] = estimate_e(20);

c) N=40N = 40N=40:

matlabCopyEdit[e_40, error_40] = estimate_e(40);

The function will display the estimated value of eee along with the corresponding true relative error for each NNN.