Problem
Examples:
Find the Fourier series of the function \( f(x)=x+\pi \) if \( -\pi
Solution
Step 1 ：Step 1: Calculate Fourier coefficients a_0, a_n and b_n
Step 2 ：Step 2: a_0 = \( \frac{1}{\pi} \int_{-\pi}^{\pi} (x + \pi) \, dx \)
Step 3 ：Step 3: a_0 = \( \frac{1}{\pi} \times \pi = 1 \)
Step 4 ：Step 4: a_n = \( \frac{1}{\pi} \int_{-\pi}^{\pi} (x + \pi) \cos(nx) \, dx \)
Step 5 ：Step 5: a_n = 0
Step 6 ：Step 6: b_n = \( \frac{1}{\pi} \int_{-\pi}^{\pi} (x + \pi) \sin(nx) \, dx \)
Step 7 ：Step 7: b_n = \( \frac{2}{n} \)
Step 8 ：Step 8: Fourier series = \( f(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos(nx) + b_n \sin(nx) \)
Step 9 ：Step 9: Fourier series = \( f(x) = 1 + \sum_{n=1}^{\infty} \frac{2}{n} \sin(nx) \)
From Solvely APP
Solution
Step 1 ：Step 1: Calculate Fourier coefficients a_0, a_n and b_n
Step 2 ：Step 2: a_0 = \( \frac{1}{\pi} \int_{-\pi}^{\pi} (x + \pi) \, dx \)
Step 3 ：Step 3: a_0 = \( \frac{1}{\pi} \times \pi = 1 \)
Step 4 ：Step 4: a_n = \( \frac{1}{\pi} \int_{-\pi}^{\pi} (x + \pi) \cos(nx) \, dx \)
Step 5 ：Step 5: a_n = 0
Step 6 ：Step 6: b_n = \( \frac{1}{\pi} \int_{-\pi}^{\pi} (x + \pi) \sin(nx) \, dx \)
Step 7 ：Step 7: b_n = \( \frac{2}{n} \)
Step 8 ：Step 8: Fourier series = \( f(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos(nx) + b_n \sin(nx) \)
Step 9 ：Step 9: Fourier series = \( f(x) = 1 + \sum_{n=1}^{\infty} \frac{2}{n} \sin(nx) \)
