Fourier Series
Definition
Building any repeating wave from pure sine waves — the mathematics behind MP3s and musical timbre.
Technical definition
A periodic f(t) = a₀/2 + Σ(aₙcos nωt + bₙsin nωt), with coefficients from orthogonality integrals; a square wave needs infinite odd harmonics.
Example
A clarinet's hollow tone is mostly odd harmonics — its Fourier series explains why it sounds different from a flute playing the same note.
Common confusion
Fourier series are for PERIODIC functions. Non-repeating signals need the Fourier transform instead — same family, different tool.