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.