This  notebook demonstrates the Gibbs phenomenon in Fourier Series.

Partial sums of the Fourier sine series of f(x) = 1 on the interval 0 < x < 1 are shown.  These are also the partial sums of the Fourier series of g(x) = 1 if x > 0 and g(x) = -1 if x < 0. on the interval -1 < x < 1.

If you fix an x value and then let the number of terms -> ∞, then the Fourier Series converges to the red function.  We say that the series converges pointwise.  But, every partial sum of the Fourier series overshoots somewhere near a jump in the red function.  The overshoot is approximately 9% of the value of the jump.  We say that the Fourier series does not converge uniformly.

Manipulate[Plot[  (∑_(n=1)^numTerms)((2(1.-(((-1))^n)))/(n π)) Sin[n π x], {x, -1.5, 1.5}, PlotRange -> {-1.3,1.3}, Prolog ->{Red , PointSize[0.015],Line[{{0,1}, {1,1}}], Line[{{-1,-1}, {0,-1}}], Line[{{1, -1},{1.5, -1}}],Line[{{-1.5, 1},{-1, 1}}], Point[{-1,0}], Point[{0,0}], Point[{1,0}]}], {{numTerms, 19, "number of terms in partial sum"}, {1,3, 5, 7, 9,19, 29,39, 49, 59, 99}}, ControlType -> Setter, SaveDefinitions -> True]