What are the nonzero sine coefficients of the square wave and how do they behave?

Conditions

• The function is the specified $2\pi$-periodic square wave.
• Coefficients are indexed by $k \ge 0$ corresponding to frequency $2k+1$.

Reasoning, step by step

1. Identify the frequency indices for nonzero coefficients (odd integers).
2. Apply the coefficient formula $4/[\pi(2k+1)]$.
3. Observe the sign of the coefficients for successive $k$.
4. Conclude that the coefficients are positive and non-alternating.

Example

The script states: 'Nonzero sine coefficients occur at odd frequencies and are positive: $4/[π(2k+1)]$. They do not alternate in sign.'

Common misconceptions

• Expecting the coefficients to alternate in sign like the alternating harmonic series.
• Assuming even harmonics have nonzero coefficients.

Watch the explanation

Connected concepts

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• How do low-order partial sums build the square wave?
• What happens at the jumps of the square wave?