ABSTRACT
To better see how the series converges towards the function, Figure 2 shows the sums of the first 1, 2, 3, and 4 terms of the series.
We had to say that virtually any periodic function can be represented as a Fourier series because in 1872, some exceptions were found: continuous functions whose Fourier series diverges from the function at certain points. (Some large-minded mathematicians consider these series convergent, but even these mathematicians boggle at a function discovered by Andrei Kolmogorov in 1923, whose Fourier series diverges everywhere.)
Actually, even when a Fourier series does converge towards its function, convergence doesn't always work as neatly as one might hope, and the man on the street might find something a bit fishy about the mathematical definition of convergence. A function like the square-wave function, which is piecewise continuous (continuous in stretches, between the discontinuities) and piecewise differentiable (the parts between the discontinuities are very smooth) has, as Gustave Lejeune Dirichlet proved, a pointwise convergent Fourier series.
