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The function \( F(x) \) is a piecewise function defined as:

\[
F(x) = \begin{cases} 
3, & 0 < x < 5 \\
-3, & -5 < x < 0 
\end{cases}
\]

### To compute the Fourier Series

The Fourier series for a periodic function is given by:

\[
F(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos \left( \frac{2\pi n x}{T} \right) + b_n \sin \left( \frac{2\pi n x}{T} \right) \right)
\]

Where \( T \) is the period of the function, and the coefficients \( a_0 \), \( a_n \), and \( b_n \) are determined as follows:

- \( a_0 \) (DC term) is calculated by:
\[
a_0 = \frac{2}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} F(x) \, dx
\]
- \( a_n \) (cosine coefficients) are calculated by:
\[
a_n = \frac{2}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} F(x) \cos \left( \frac{2\pi n x}{T} \right) \, dx
\]
- \( b_n \) (sine coefficients) are calculated by:
\[
b_n = \frac{2}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} F(x) \sin \left( \frac{2\pi n x}{T} \right) \, dx
\]

### Steps:
1. **Determine the period \( T \)**: 
   Since the function is defined in the interval \( -5 < x < 5 \), the period \( T = 10 \).

2. **Calculate \( a_0 \) (DC term)**: 
   The average value of the function over one period.

3. **Calculate \( a_n \) and \( b_n \)** using the above formulas.

Shall we proceed with calculating the Fourier series terms in detail?

Explore the detailed steps to compute the Fourier series for a piecewise function defined on the interval [-5, 5]. This guide covers key concepts like the Fourier series formula, DC term, and the calculation of sine and cosine coefficients. Suitable for college-level students.

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