% Define the function to be approximated

syms f(x)

f(x) = sin(x);

% Set the number of terms in the Fourier series

N = 10;

% Define the range of x values to plot

xmin = -pi;

xmax = pi;

dx = 0.01;

% Calculate the Fourier series coefficients

syms n

a0 = simplify(int(f(x), x, xmin, xmax)/(2*pi));

an = simplify(int(f(x)*cos(n*x), x, xmin, xmax)/pi);

bn = simplify(int(f(x)*sin(n*x), x, xmin, xmax)/pi);

% Calculate the Fourier series approximation

f_series = a0;

for n = 1:N

f_series = f_series + an*cos(n*x) + bn*sin(n*x);

end

% Plot the original function and the Fourier series approximation

x = xmin:dx:xmax;

figure;

plot(x, double(f(x)), 'b-');

hold on;

plot(x, double(subs(f_series, x)), 'r--');matlab:matlab.internal.language.introspective.errorDocCallback('symengine')

title(sprintf('Fourier series approximation of %s with %d terms', char(f(x)), N));

xlabel('x');

ylabel('f(x)');

legend('Original function', 'Fourier series approximation');

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