1. The problem: Understand what a Fourier series is and how it represents periodic functions. 2. A Fourier series expresses a periodic function $f(x)$ as an infinite sum of sines and cosines: $$f(x) = a_0 + \sum_{n=1}^\infty \left(a_n \cos(nx) + b_n \sin(nx)\right)$$ where $a_0$, $a_n$, and $b_n$ are coefficients. 3. The coefficients are calculated using integrals over one period $[-\pi, \pi]$: $$a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \, dx$$ $$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx$$ $$b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx$$ 4. Important rules: - The function must be periodic and integrable over the interval. - The series converges to $f(x)$ at points where $f$ is continuous. - At discontinuities, it converges to the average of left and right limits. 5. Fourier series allow us to analyze complex periodic signals by breaking them into simple oscillations. This explanation covers the basics of Fourier series and how to compute them.