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Find the Fourier series of the given function , which is
assumed to have the period . Show the details of your
work. Sketch or graph the partial sums up to that including cos 5x and sin 5x

The plot compares the **original function** (red dashed line) with the **partial Fourier sum** up to \( \sin(5x) \) (blue line). As seen, the Fourier series approximates the original function, but due to the limited number of terms, there are some oscillations (known as **Gibbs phenomenon**) near the discontinuities.

### Summary of the Fourier Series:
The Fourier series for the given function (up to \( \sin(5x) \)) is:

\[
f(x) \approx \frac{4}{\pi} \sin(x) + \frac{4}{3\pi} \sin(3x) + \frac{4}{5\pi} \sin(5x)
\]

Would you like further details or additional questions on this topic?

### Related Questions:
1. What would happen if we included more terms in the Fourier series?
2. Can the Fourier series approximate a discontinuous function exactly?
3. What is the physical interpretation of the Fourier coefficients?
4. How does symmetry simplify Fourier series calculations?
5. What is the Gibbs phenomenon, and how does it affect Fourier series?

**Tip:** Including more terms in a Fourier series improves the accuracy of approximation, especially near discontinuities, but cannot fully eliminate oscillations at these points.

Find the Fourier series of the given function, which is assumed to have the period. Show the details of your work. Sketch or graph the partial sums up to that including cos 5x and sin 5x.

Explore the Fourier series approximation for a given periodic function. The solution includes the partial sum approximation up to sin(5x) and explains concepts like the Gibbs phenomenon. The series approximation covers essential Fourier components for undergraduate students.

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