a) Consider the periodic function f(x) defined by : f(x)= ) 0,&-pi lt xlt 0 x,&0leqslant xlt pi i.Sketch the function in the interval [-3pi ,3pi ] ii. Find the Fourier series expansion for f(x) Hence sum the series: 1+(1)/(3^2)+(1)/(5^2)+(1)/(7^2)+...

Let's correct and complete the Fourier series expansion for \( f(x) \) and sum the series.### i. Sketch the function in the interval The function \( f(x) \) is defined as: This means \( f(x) \) is zero for and \( f(x) = x \) for . To sketch this:- For , \( f(x) = 0 \).- For , \( f(x) = 0 \).- For , \( f(x) = x \).- For , \( f(x) = 0 \).### ii. Find the Fourier series expansion for \( f(x) \)The Fourier series expansion of a periodic function \( f(x) \) with period is given by: Where the coefficients and are given by: For \( f(x) \) defined as: #### Compute : Since \( f(x) = 0 \) for : #### Compute : Since \( f(x) = 0 \) for : #### Compute :\[ b_n = \frac{1}{\pi}