What are the nonzero sine coefficients of the standard odd square wave and do they alternate in sign?

Conditions

• The square wave is defined as 1 on $(0, \pi)$ and -1 on $(-\pi, 0)$.
• Only odd harmonics are considered.

Reasoning, step by step

1. Determine which frequencies have nonzero coefficients: only odd integers.
2. Calculate the magnitude of the coefficient for the $k$-th odd harmonic.
3. Verify the sign of the coefficients using the integration formula for sine terms.
4. Observe that the result $\frac{4}{\pi(2k+1)}$ is strictly positive for all $k \ge 0$.

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

• Assuming that alternating signs are typical for square waves; this specific phase alignment yields positive coefficients.
• Including even harmonics in the sum; their coefficients are exactly zero due to symmetry.

Watch the explanation

Explore next

• How does shifting the square wave horizontally change the signs of these coefficients?
• Why do even harmonics vanish completely?