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 a 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 by: $$a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \, dx$$ $$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx$$ $$b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx$$ 4. Important rules: - The function must be periodic with period $2\pi$ (or can be adjusted for other periods). - 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. Explanation: The Fourier series breaks down any periodic function into simple waves (sines and cosines) of different frequencies and amplitudes. 6. This is useful in signal processing, physics, and engineering to analyze complex waveforms. 7. Example: For $f(x) = x$ on $[-\pi, \pi]$, the Fourier series is: $$f(x) = 2 \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin(nx)$$ which shows only sine terms because $f(x)$ is odd. This completes the explanation of the Fourier series.