How do I solve Fourier series problems step by step?

A good Fourier series workflow is almost mechanical once you know what to check. 1. Identify the interval. Most introductory problems use $[-\pi,\pi]$ or $[0,L]$. 2. Check symmetry. If $f(x)$ is even, then all sine coefficients vanish: $$b_n=0.$$ If $f(x)$ is odd, then all cosine coefficients vanish: $$a_n=0.$$ 3. Write the coefficient formulas before integrating: $$a_n=\frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\cos(nx)\,dx,$$ $$b_n=\frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\sin(nx)\,dx.$$ 4. Compute only the coefficients that can be nonzero. 5. Substitute them back into the series. The common mistake is to start integrating immediately. Look for symmetry first. It can cut the work in half.