Fourier Series Calculation:
$$a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} x \, dx = \frac{x^2}{2\pi} \Big|_{-\pi}^{\pi} = 0$$
$$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x \cos(nx) \, dx = 0$$
$$b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x \sin(nx) \, dx = -\frac{2}{n} (-1)^n$$
Fourier Series Representation:
$$f(x) = -2 \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \sin(nx)$$
First few terms:
$$f(x) = 2\sin(x) - \sin(2x) + \frac{2}{3}\sin(3x) + \dots$$
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