Examples:
Find the Fourier series of the function \( f(x)=x+\pi \) if \( -\pi< x< \pi \) and \( f(x+2 \pi)=f(x) \).
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 = \( f(x) = 1 + \sum_{n=1}^{\infty} \frac{2}{n} \sin(nx) \)