(Solution) - The partial sums sn x of a Fourier series show oscillations -(2025 Original AI-Free Solution)

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The partial sums sn(x) of a Fourier series show oscillations near a discontinuity point. These oscillations do not disappear as n increases but instead become sharp "spikes". They were explained mathematically by J.W Gibbs3. Graph sn(x) in Prob. 10. When n = 50, say, you will see those oscillations quite distinctly. Consider other Fourier series of your choice in a similar way. Compare.