The amplitudes of the optimal sine waves follow a prescription that Fourier discovered: Triangle wave = sinx − 1⁄9sin3x + 1⁄25sin5x − 1⁄49sin7x + · · · . This infinite sum is called the Fourier series for the triangle wave. Notice the cool numerical patterns in it. Only odd frequencies 1, 3, 5, 7, . . . appear in the sine waves, and their corresponding amplitudes are the inverse squares of the odd numbers with alternating plus and minus signs. Unfortunately, I can’t easily explain why this prescription works; we would have to plow through too much nitty-gritty calculus to see where those magic
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