Let's solve each part step-by-step. Recall the Fourier series for a function \( f(x) \) with period \( 2\pi \) is: \[ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos(nx) + b_n \sin(nx) \right] \] where \[ a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\,dx \] \[ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx)\,dx \] \[ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx)\,dx \] --- ### **a) \( f(x) = |x| \), \( -\pi < x < \pi \)** \( f(x) \) is **even**, so \( b_n = 0 \). #### **Find \( a_0 \):** \[ a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} |x| dx = \frac{2}{\pi} \int_{0}^{\pi} x dx = \frac{2}{\pi} \left[ \frac{x^2}{2} \right]_0^{\pi} = \frac{2}{\pi} \cdot \frac{\pi^2}{2} = \pi \] #### **Find \( a_n \):** \[ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} |x| \cos(nx) dx \] Since \( |x| \) and \( \cos(nx) \) are even: \[ a_n = \frac{2}{\pi} \int_0^{\pi} x \cos(nx) dx \] Integrate by parts: Let \( u = x, dv = \cos(nx)dx \Rightarrow du = dx, v = \frac{\sin(nx)}{n} \) \[ \int x \cos(nx) dx = x \frac{\sin(nx)}{n} - \int \frac{\sin(nx)}{n} dx = x \frac{\sin(nx)}{n} + \frac{\cos(nx)}{n^2} \] So, \[ a_n = \frac{2}{\pi} \left[ x \frac{\sin(nx)}{n} + \frac{\cos(nx)}{n^2} \right]_0^{\pi} \] Calculate at bounds: At \( x = \pi \): - \( \sin(n\pi) = 0 \) - \( \cos(n\pi) = (-1)^n \) At \( x = 0 \): - \( \sin(0) = 0 \) - \( \cos(0) = 1 \) Thus, \[ a_n = \frac{2}{\pi} \left( 0 + \frac{(-1)^n}{n^2} - 0 - \frac{1}{n^2} \right) = \frac{2}{\pi} \left( \frac{(-1)^n - 1}{n^2} \right) \] So, for **odd \( n \): (-1)^n - 1 = -2**, for **even \( n \): 0** \[ a_n = \begin{cases} -\frac{4}{\pi n^2}, & n \text{ odd} \\ 0, & n \text{ even} \end{cases} \] #### **Final Fourier Series:** \[ |x| = \frac{\pi}{2} - \frac{4}{\pi} \sum_{\substack{n=1 \\ n \text{ odd}}}^\infty \frac{\cos(nx)}{n^2} \] --- ### **b) \( f(x) = x^2 \), \( -\pi < x < \pi \)** \( f(x) \) is **even**, so \( b_n = 0 \). #### **Find \( a_0 \):** \[ a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 dx = \frac{2}{\pi} \int_0^{\pi} x^2 dx = \frac{2}{\pi} \left[ \frac{x^3}{3} \right]_0^{\pi} = \frac{2}{\pi} \cdot \frac{\pi^3}{3} = \frac{2 \pi^2}{3} \] #### **Find \( a_n \):** \[ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \cos(nx) dx = \frac{2}{\pi} \int_0^{\pi} x^2 \cos(nx) dx \] Integrate by parts twice (result standard): \[ \int x^2 \cos(nx) dx = \frac{x^2 \sin(nx)}{n} + \frac{2x \cos(nx)}{n^2} - \frac{2 \sin(nx)}{n^3} \] So, \[ a_n = \frac{2}{\pi} \left[ \frac{x^2 \sin(nx)}{n} + \frac{2x \cos(nx)}{n^2} - \frac{2\sin(nx)}{n^3} \right]_0^{\pi} \] At \( x = \pi \): - \( \sin(n\pi) = 0 \) - \( \cos(n\pi) = (-1)^n \) At \( x = 0 \): - \( \sin(0) = 0 \) - \( \cos(0) = 1 \) So, \[ a_n = \frac{2}{\pi} \left( 0 + \frac{2\pi(-1)^n}{n^2} - 0 - (0 + \frac{0}{n^2} - 0) \right) = \frac{4(-1)^n}{n^2} \] #### **Final Fourier Series:** \[ x^2 = \frac{\pi^2}{3} + 4 \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} \cos(nx) \] --- ### **c)** \[ f(x) = \begin{cases} 0, & -\pi < x < 0 \\ 1, & 0 < x < \pi \end{cases} \] This is a half-range step function. #### **Find \( a_0 \):** \[ a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) dx = \frac{1}{\pi} \int_{0}^{\pi} 1 dx = \frac{1}{\pi} \cdot \pi = 1 \] #### **Find \( a_n \):** \[ a_n = \frac{1}{\pi} \int_{0}^{\pi} \cos(nx) dx = \frac{1}{\pi} \left[ \frac{\sin(nx)}{n} \right]_0^{\pi} = \frac{1}{\pi} \left( \frac{\sin(n\pi)}{n} - 0 \right) = 0 \] Because \( \sin(n\pi) = 0 \). #### **Find \( b_n \):** \[ b_n = \frac{1}{\pi} \int_{0}^{\pi} \sin(nx) dx = \frac{1}{\pi} \left[ -\frac{\cos(nx)}{n} \right]_0^{\pi} = \frac{1}{\pi} \left( -\frac{\cos(n\pi)}{n} + \frac{1}{n} \right) \] \[ = \frac{1}{\pi} \left( \frac{1 - (-1)^n}{n} \right) \] So for **odd \( n \): 1-(-1)^n = 2**, for **even \( n \): 0** \[ b_n = \begin{cases} \frac{2}{\pi n}, & n \text{ odd} \\ 0, & n \text{ even} \end{cases} \] #### **Final Fourier Series:** \[ f(x) = \frac{1}{2} + \frac{2}{\pi} \sum_{\substack{n=1 \\ n \text{ odd}}}^\infty \frac{\sin(nx)}{n} \] --- ### **d)** \[ f(x) = \begin{cases} 0, & -\pi < x < 0 \\ x, & 0 < x < \pi \end{cases} \] #### **Find \( a_0 \):** \[ a_0 = \frac{1}{\pi} \int_{0}^{\pi} x dx = \frac{1}{\pi} \cdot \frac{\pi^2}{2} = \frac{\pi}{2} \] #### **Find \( a_n \):** \[ a_n = \frac{1}{\pi} \int_{0}^{\pi} x \cos(nx) dx \] Integrate by parts (\( u = x, dv = \cos(nx) dx \)): \[ = \frac{1}{\pi} \left[ x \frac{\sin(nx)}{n} \Big|_0^{\pi} - \int_0^{\pi} \frac{\sin(nx)}{n} dx \right] \] \[ = \frac{1}{\pi} \left( \frac{\pi \sin(n\pi)}{n} - 0 - \frac{1}{n} \int_0^{\pi} \sin(nx) dx \right) \] \[ \int_0^{\pi} \sin(nx) dx = \left[ -\frac{\cos(nx)}{n} \right]_0^{\pi} = -\frac{\cos(n\pi) - 1}{n} \] So, \[ a_n = \frac{1}{\pi} \left( 0 + \frac{1}{n^2} [\cos(n\pi) - 1]\right) = \frac{1}{\pi n^2} [(-1)^n - 1] \] So for odd \( n \): \( (-1)^n - 1 = -2 \), for even \( n \): \( 0 \) \[ a_n = \begin{cases} -\frac{2}{\pi n^2}, & n \text{ odd} \\ 0, & n \text{ even} \end{cases} \] #### **Find \( b_n \):** \[ b_n = \frac{1}{\pi} \int_{0}^{\pi} x \sin(nx) dx \] Integrate by parts (\( u = x, dv = \sin(nx) dx \)): \[ = \frac{1}{\pi} \left( -x \frac{\cos(nx)}{n} \Big|_0^{\pi} + \int_0^{\pi} \frac{\cos(nx)}{n} dx \right) \] \[ = \frac{1}{\pi} \left( -\frac{\pi \cos(n\pi)}{n} + 0 + \frac{1}{n} \int_0^{\pi} \cos(nx) dx \right) \] \[ \int_0^{\pi} \cos(nx) dx = \left[ \frac{\sin(nx)}{n} \right]_0^{\pi} = 0 \] So, \[ b_n = -\frac{\cos(n\pi)}{n} = -\frac{(-1)^n}{n} \] #### **Final Fourier Series:** \[ f(x) = \frac{\pi}{4} - \frac{2}{\pi} \sum_{\substack{n=1 \\ n \text{ odd}}}^\infty \frac{\cos(nx)}{n^2} - \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \sin(nx) \] --- ## **Summary Table** | Part | \( f(x) \) | Fourier Series | |------|------------|---------------| | (a) | \( |x| \) | \( \frac{\pi}{2} - \frac{4}{\pi}\sum_{\substack{n=1 \\ n\,\text{odd}}}^{\infty} \frac{\cos(nx)}{n^2} \) | | (b) | \( x^2 \) | \( \frac{\pi^2}{3} + 4\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}\cos(nx) \) | | (c) | step | \( \frac{1}{2} + \frac{2}{\pi} \sum_{\substack{n=1 \\ n \text{ odd}}}^{\infty} \frac{\sin(nx)}{n} \) | | (d) | ramp | \( \frac{\pi}{4} - \frac{2}{\pi} \sum_{\substack

a) \( |x| = \frac{\pi}{2} - \frac{4}{\pi} \sum_{\substack{n=1,\, n\,\text{odd}}}^\infty \frac{\cos(nx)}{n^2} \) b) \( x^2 = \frac{\pi^2}{3} + 4 \sum_{n=1}^\infty \frac{(-1)^n}{n^2} \cos(nx) \) c) \( f(x) = \frac{1}{2} + \frac{2}{\pi} \sum_{\substack{n=1,\, n\,\text{odd}}}^\infty \frac{\sin(nx)}{n} \) d) \( f(x) = \frac{\pi}{4} - \frac{2}{\pi} \sum_{\substack{n=1,\, n\,\text{odd}}}^\infty \frac{\cos(nx)}{n^2} - \sum_{n=1}^\infty \frac{(-1)^n}{n} \sin(nx) \)