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The question asks for the Fourier series of a given function \( f(x) \). The function is defined as:

\[
f(x) = 
\begin{cases}
x - \lfloor x \rfloor - \frac{1}{2}, & \text{if } x \text{ is not an integer}, \\
0, & \text{if } x \text{ is an integer},
\end{cases}
\]
where \( \lfloor x \rfloor \) is the greatest integer function (also known as the floor function), within the interval \( \left[ -\frac{1}{2}, \frac{1}{2} \right] \).

Given the periodicity of \( f(x) \) with \( f(x+1) = f(x) \), this is a typical setup for finding the Fourier series of a periodic function.

### Fourier Series
For a periodic function \( f(x) \) with period \( 2L \), the Fourier series can be expressed as:

\[
f(x) = \sum_{n=1}^{\infty} b_n \sin\left(\frac{n\pi x}{L}\right),
\]
if the function is odd (as \( f(x) \) is in this case). Here, the coefficients \( b_n \) are given by:

\[
b_n = \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) \, dx.
\]

### For this function:
- The function is defined to have period 1, so \( L = \frac{1}{2} \).
- The function is odd, so we only expect sine terms in the Fourier series expansion.

Looking at the choices, the correct Fourier series representation should be:

\[
f(x) = \frac{2}{\pi} \sum_{n=1}^{\infty} \frac{\sin(n\pi x)}{n}.
\]

Thus, the correct answer is the second option:

\[
\boxed{f(x) = \frac{2}{\pi} \sum_{n=1}^{\infty} \frac{\sin(n\pi x)}{n}}.
\]

Would you like further details or clarification on this? Here are some related questions:

1. What is the general process for finding the Fourier coefficients for a periodic function?
2. Why are there only sine terms in the Fourier series of this function?
3. How does the symmetry of a function influence its Fourier series expansion?
4. What is the significance of the greatest integer function \( \lfloor x \rfloor \) in this context?
5. How do the Fourier coefficients \( b_n \) relate to the function's behavior at different points?

**Tip**: Always check the symmetry of a function (odd or even) before deciding which terms (sine or cosine) will appear in its Fourier series.

The Fourier series of the function f(x) defined as x - [x] - 1/2 for non-integer x and x for integer x, where [x] is the greatest integer function in the interval [-1/2, 1/2].

Explore the Fourier series representation of a function defined using the greatest integer function. Learn how to expand the periodic function f(x) = x - [x] - 1/2 with respect to Fourier coefficients. Suitable for undergraduate mathematics students.

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