how to approximate e^0.5 using a taylor series in matlab

To approximate e^0.5 using a Taylor series in MATLAB, we can use the formula:

main.m
e^x = sum(n=0 to infinity) (x^n / n!)
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Since we want to approximate e^0.5, we'll substitute x = 0.5:

main.m
e^0.5 = sum(n=0 to infinity) (0.5^n / n!)
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To approximate this infinite series, we can truncate it after a certain number of terms. Let's say we want to approximate e^0.5 to within an error of 0.0001. We can use the following MATLAB code:

main.m
x = 0.5;
approx = 1;
error = 1;
n = 1;
while error > 0.0001
    approx = approx + x^n / factorial(n);
    error = abs(exp(x) - approx);
    n = n + 1;
end
disp(approx)
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In this code, we initialize the approximation to 1, the error to 1, and the number of terms to 1. We then enter a loop that continues until the error is less than 0.0001. In each iteration of the loop, we add the next term of the Taylor series to the approximation, calculate the error as the absolute value of the difference between the true value of e^0.5 and our approximation, and increment n. Finally, we display the resulting approximation.

Note that this method of approximating e^0.5 using a Taylor series is not the most efficient, especially for larger values of x. There are more efficient algorithms for approximating the exponential function, such as the Taylor series with Horner's rule, the CORDIC algorithm, or rational approximations.