Abstract
When a Fourier series is used to approximate a periodic and piecewise smooth function with a jump discontinuity, an overshoot at the discontinuity occurs and is called Gibbs phenomenon. For explaining understandably and systematically this Gibbs phenomenon from the educational point of view , the representation method is proposed using an accordion-like folding convergence of the extremum values for the partial sums of its Fourier series at the discontinuity. Using an integration by parts to obtain Fourier coefficients and rearranging its Fourier series, the expansions are devised to show explicitly the overshoots at the discontinuities for any functions with jump discontinuities.
