Examples:
Find the Fourier series of the function $f (x) = x + π$ if $- π < x < π$ and $f (x + 2 π) = f (x)$ .
Solution:

Answer

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Answer

Step 9: Fourier series = $f (x) = 1 + \sum_{n = 1}^{\infty} \frac{2}{n} \sin (n x)$

Steps

Step 1 ：Step 1: Calculate Fourier coefficients a_0, a_n and b_n

Step 2 ：Step 2: a_0 = $\frac{1}{π} \int_{- π}^{π} (x + π) d x$

Step 3 ：Step 3: a_0 = $\frac{1}{π} \times π = 1$

Step 4 ：Step 4: a_n = $\frac{1}{π} \int_{- π}^{π} (x + π) \cos (n x) d x$

Step 5 ：Step 5: a_n = 0

Step 6 ：Step 6: b_n = $\frac{1}{π} \int_{- π}^{π} (x + π) \sin (n x) d x$

Step 7 ：Step 7: b_n = $\frac{2}{n}$

Step 8 ：Step 8: Fourier series = $f (x) = a_{0} + \sum_{n = 1}^{\infty} a_{n} \cos (n x) + b_{n} \sin (n x)$

Step 9 ：Step 9: Fourier series = $f (x) = 1 + \sum_{n = 1}^{\infty} \frac{2}{n} \sin (n x)$