Question

Find the Fourier series of the periodic function (x) = x + ,

− < x < in the simplest form. Then, sketch the graph of
the periodic function (x) for −3 &lt

Answer 1

To find the Fourier series of the function (x) = x + ,− < x < , determine the coefficients of the sine and cosine terms. The graph can be sketched by plotting the points over one period and repeating the pattern.

Step-by-step explanation:

The Fourier series of a periodic function is a way to represent the original function as a sum of sine and cosine functions. To find the Fourier series of the function (x) = x + ,− < x < , we need to determine the coefficients of the sine and cosine terms.

Since the function is odd, all the cosine terms will be zero. The sine terms can be found using the formula:

a_n = (2/L) ∫(−L to L) f(x) sin(nπx/L) dx

where L is the period of the function.

Once the coefficients are found, the Fourier series can be written as:

f(x) = a_0 + Σ(a_n sin(nπx/L))

Where Σ denotes summation from n=1 to infinity.

To sketch the graph of the periodic function, plot the points of the function over one period and repeat the pattern for the entire range of x-values.