How does the Gibbs overshoot behave as more terms are added to the square wave series?

Conditions

• The series is the Fourier expansion of a function with a jump discontinuity.
• The number of terms $N$ approaches infinity.

Reasoning, step by step

1. Observe the partial sums near the jump.
2. Note the narrowing of the region containing the overshoot.
3. Measure the peak height of the overshoot relative to the flat level.
4. Observe that the peak height remains nonzero as $N$ increases.
5. Conclude that uniform convergence fails across the jump.

Example

The script states: 'Nearby overshoot narrows in location but its relative peak does not vanish. This distinguishes pointwise convergence from uniform convergence across the jumps.'

Common misconceptions

• Believing that the overshoot disappears completely with enough terms.
• Thinking that the narrowing location implies the peak height also decreases.

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Connected concepts

Explore next

• Why does the series converge pointwise but not uniformly?
• What is the significance of the midpoint value at the jump?