MATLAB required

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MATLAB required: Write a While loop (only the while loop) to compute Euler’s number otherwise known as the natural log e. This is a convergence problem. Euler’s number is expressed as e = \sum_{n=0}^{\infty} \frac{1}{n!} This must not go to infinite so stop the loop when the term becomes < .00000001. Think about the ideas in Taylor Series. Note n! gets large very fast.

The Correct Answer and Explanation is:

The formula for Euler’s number (e) is given by the Taylor series expansion:e=∑n=0∞1n!e = \sum_{n=0}^{\infty} \frac{1}{n!}e=n=0∑∞​n!1​

In MATLAB, we can compute Euler’s number using a while loop. The loop will stop when the term becomes less than 10−810^{-8}10−8, or 0.000000010.000000010.00000001, as specified.

Here’s how you can write the while loop:

matlabCopyEditn = 0;  % Initialize the counter
term = 1;  % The first term (1/0!) is 1
e_approx = 0;  % Variable to hold the approximation of e

while term > 1e-8  % Continue until the term is less than 0.00000001
    e_approx = e_approx + term;  % Add the current term to the sum
    n = n + 1;  % Increment the counter
    term = 1 / factorial(n);  % Calculate the next term in the series
end

disp(e_approx)  % Display the computed value of e

Explanation:

  1. Initialization:
    • n starts at 0 because the series begins with 10!\frac{1}{0!}0!1​, which is 1.
    • term is initialized to 1, the first term in the Taylor series expansion.
    • e_approx starts at 0 and will accumulate the sum of terms.
  2. While loop condition:
    • The loop runs as long as term is greater than 10−810^{-8}10−8, ensuring we keep adding significant terms and stop when they become negligible.
  3. Accumulation:
    • Inside the loop, the current term is added to e_approx, which gradually converges to the value of eee.
    • After each iteration, the counter n is incremented, and term is recalculated as 1n!\frac{1}{n!}n!1​.
  4. Stopping criterion:
    • The loop stops once the term being added becomes smaller than 10−810^{-8}10−8, which ensures the desired precision.
  5. Factorial growth:
    • Factorial grows extremely fast, which is why the terms become very small quickly. This allows us to stop the computation early without needing to sum an infinite number of terms.

Running this loop will give an approximation of eee that converges to the true value with good precision.

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