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Let's solve this step by step to obtain the Fourier series expansion for the given function. We are given a piecewise function with period $\pi$: $$ f(x) = \begin{cases} \frac{4}{\pi} x & \text{for } 0 \le x < \frac{\pi}{2} \\[2mm] -\frac{4}{\pi} x & \text{for } -\frac{\pi}{2} \le x \le 0 \end{cases} $$ --- ### Step 1: Identify type of function * The function is odd, because $f(-x) = -f(x)$. * For an odd function, the Fourier series contains only sine terms: $$ f(x) = \sum_{n=1}^{\infty} b_n \sin(nx) $$ where $$ b_n = \frac{2}{\pi} \int_{-\pi/2}^{\pi/2} f(x) \sin(nx) \, dx _________ ________ _____ _________ __________ _________ _____ ______ ________ ________ ____ ___.
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