The function shown in the image appears to be a periodic triangle wave function, oscillating between $-1$ and $1$, with period $2\pi$. The Fourier series of a periodic function can be expressed as:

$f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left(a_n \cos(nt) + b_n \sin(nt)\right)$

To find the Fourier series for this function, we need to calculate the Fourier coefficients $a_0$, $a_n$, and $b_n$.

1. $a_0$ (DC component): $a_0 = \frac{1}{T} \int_{0}^{T} f(t) \, dt$ where $T = 2\pi$ is the period.

2. $a_n$ (cosine coefficients): $a_n = \frac{2}{T} \int_{0}^{T} f(t) \cos(nt) \, dt$

3. $b_n$ (sine coefficients): $b_n = \frac{2}{T} \int_{0}^{T} f(t) \sin(nt) \, dt$

Now, looking at the function's symmetry, we can infer that it is an odd function since $f(-t) = -f(t)$. For odd functions, the cosine terms $a_n$ vanish, and the Fourier series will consist only of sine terms. Thus, $a_n = 0$.

Steps to Compute the Fourier Series:

1. The function $f(t)$ is odd, meaning we only need to compute the sine coefficients $b_n$.
2. The function seems to be piecewise linear with alternating slopes. The integral for each piece can be split over the intervals where the function has different linear expressions.

Would you like me to continue with a detailed calculation of the Fourier coefficients $b_n$, or is there any particular part you'd like to explore first?

Relative Questions:

1. What is the significance of using Fourier series for periodic functions?
2. How does the symmetry of the function affect the Fourier series components?
3. Can you explain why the cosine terms vanish for odd functions?
4. How do the periods and amplitude affect the Fourier coefficients?
5. What are some applications of Fourier series in signal processing?

Tip:

For functions with clear symmetry (odd or even), simplifying the Fourier series by eliminating unnecessary terms (either sine or cosine) can significantly reduce the work needed.