Fourier coefficient
If a Fourier series is a recipe that builds a function from pure tones, the Fourier coefficients are the amounts — how much of each tone you pour in. The coefficient at frequency n measures how strongly the function ‘resonates’ with the wave of that frequency.
The extraction rule is a projection: integrate the function against the matching wave. In complex form, c_n = (1/(2*pi)) integral over [-pi, pi] of f(x) e^{-i n x} dx. In real form, a_n = (1/pi) integral of f(x) cos(n x) dx and b_n = (1/pi) integral of f(x) sin(n x) dx. This works because the trigonometric functions form an orthogonal system: integrating cos(n x) against cos(m x) gives zero unless n = m, so every other tone ‘cancels out’ and only the one you are testing for survives.
Coefficients encode smoothness in their rate of decay: the smoother f is, the faster c_n tends to 0. If f is merely integrable, c_n -> 0 (the Riemann-Lebesgue lemma); if f is k-times continuously differentiable and periodic, then n^k c_n -> 0. Conversely, slow decay signals roughness — a jump discontinuity forces c_n to decay only like 1/n.
For f(x) = x on (-pi, pi), extended periodically, b_n = (1/pi) integral of x sin(n x) dx = (-1)^{n+1} (2/n) and all a_n = 0. So the coefficients decay like 1/n, matching the jump that the periodic extension has at x = pi.
Odd function -> only sine terms; the 1/n decay is the fingerprint of a discontinuity.