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  "body": "\n## Example\n\nLet $f$ be the $2\\pi$-periodic function with $f(x) = \\frac{\\pi - x}{2}$ for $0 \u003c x \u003c 2\\pi$ and $f(0) = 0$.\nIts graph is a sawtooth: it falls linearly from $\\frac{\\pi}{2}$ to $-\\frac{\\pi}{2}$ on each period and jumps back up at the multiples of $2\\pi$.\nIt is odd, so its cosine coefficients vanish, and integration by parts gives for $n \\ge 1$\n$$b_n = \\frac{1}{\\pi} \\int_0^{2\\pi} \\frac{\\pi - x}{2} \\sin nx \\, dx = \\frac{1}{\\pi} \\cdot \\frac{1}{2} \\cdot \\frac{2\\pi}{n} = \\frac{1}{n}.$$\nSo its [[Fourier series|G12KXDHPPS]] is\n$$\\sum_{n=1}^{\\infty} \\frac{\\sin nx}{n}.$$\nThe function has bounded variation, so by the [[Dirichlet conditions|75ZQW12WY0]] the series converges to $\\frac{\\pi - x}{2}$ for $0 \u003c x \u003c 2\\pi$, and to $0$ at the [[jumps|004FH91BN5]] $x \\in 2\\pi\\Z$.\n\n## Remarks\n\nAt $x = \\pi/2$ the series gives $\\frac{\\pi}{4} = 1 - \\frac{1}{3} + \\frac{1}{5} - \\cdots$, the Leibniz formula.\nThe coefficients $\\frac{1}{n}$ are square summable but not summable, as expected for a function with a jump.\nIntegrating the series term by term from $0$ to $x$ gives $\\sum_{n \\ge 1} \\frac{1 - \\cos nx}{n^2} = \\frac{\\pi x}{2} - \\frac{x^2}{4}$ for $0 \\le x \\le 2\\pi$, and at $x = \\pi$ this yields $\\sum_{n \\text{ odd}} \\frac{2}{n^2} = \\frac{\\pi^2}{4}$.\n\n## Explanations\n\n### L1 human\n\nThe sawtooth wave ramps down steadily and then snaps back, like the sweep of an old television beam.\nIt contains every harmonic, the $n$-th one with amplitude $\\frac{1}{n}$.\nBecause of the snap back, the amplitudes shrink slowly, and many harmonics are needed for a good picture.\n",
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