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The problem is asking for the **Fourier series expansion** of a periodic function \( f(x) \), defined over the interval \(-\pi < x < \pi\), with the condition that \( f(x+2\pi) = f(x) \), meaning the function is periodic with period \( 2\pi \).

### General Fourier Series Formulation

The Fourier series of a periodic function \( f(x) \) with period \( 2\pi \) is given by:
\[
f(x) = a_0 + \sum_{n=1}^\infty \left[ a_n \cos(nx) + b_n \sin(nx) \right]
\]
where the coefficients are defined as:
1. \( a_0 = \frac{1}{2\pi} \int_{-\pi}^\pi f(x) \, dx \)
2. \( a_n = \frac{1}{\pi} \int_{-\pi}^\pi f(x) \cos(nx) \, dx \)
3. \( b_n = \frac{1}{\pi} \int_{-\pi}^\pi f(x) \sin(nx) \, dx \)

### Steps to Solve
1. **Compute \( a_0 \):**
   - \( a_0 \) is the average value of \( f(x) \) over one period.
2. **Compute \( a_n \):**
   - These coefficients measure the cosine contributions to \( f(x) \).
3. **Compute \( b_n \):**
   - These coefficients measure the sine contributions to \( f(x) \).

### Required Input
For a complete solution, the function \( f(x) \) must be explicitly provided (e.g., \( f(x) = x \), \( f(x) = |x| \), etc.).

Would you like to proceed with a specific function for \( f(x) \), or should I provide a general explanation? Let me know! 

### 5 Questions for You:
1. Do you have a specific definition for \( f(x) \) within \(-\pi \leq x \leq \pi \)?
2. Should I demonstrate the solution using a common example, like \( f(x) = x \)?
3. Are you familiar with the integration techniques needed for calculating the coefficients?
4. Would you like a step-by-step explanation for each coefficient (\( a_0, a_n, b_n \))?
5. Should I include a graph of the Fourier series approximation?

### Tip:
Always visualize the original function and its Fourier series approximation to better understand how the series converges to the function over its domain.

Find the Fourier series expansion of the periodic function f(x), defined as -π < x < π and f(x + 2π) = f(x).

Learn how to compute the Fourier series expansion of a periodic function f(x), defined for -π < x < π and satisfying f(x + 2π) = f(x). This tutorial covers Fourier coefficients (a_0, a_n, b_n), integration techniques, and the general Fourier series formula.

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