f(x)=(a_(0))/(2)+sum _(n=1)^infty (a_(n)cos(npi x)/(L)+b_(n)sin(npi x)/(L)) f(x)=(3)/(2)+sum _(n=1)^infty (0times cos(npi x)/(5)+{ (3(1-cosnpi ))/(npi ))sin(npi x)/(5)) f(x)=(3)/(2)+(6)/(pi )(sin(pi x)/(5)+(1)/(3)sin(3pi x)/(5)+(1)/(5)sin(5pi x)/(5)+... )

The given function is a Fourier series representation of a periodic function. The Fourier series is a way to represent a periodic function as an infinite sum of sines and cosines. In this case, the function is represented as a sum of sines with different frequencies and amplitudes.The coefficients , , and are constants that determine the shape of the function. The variable represents the independent variable, and is the length of the period of the function.In the given function, the coefficient is , which represents the average value of the function over one period. The coefficient is 0, which means that the cosine term is not present in the function. The coefficient is , which represents the amplitude of the sine term at each frequency.The function can be simplified by factoring out the common factor of from each term in the sum. This gives us the final expression for the function: This expression represents the function as an infinite sum of sines with different frequencies and amplitudes. The coefficients of the sines are determined by the Fourier coefficients , which are calculated based on the original function.