{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nLet $f$ be the $2\\pi$-periodic function with $f(x) = x$ for $-\\pi \u003c x \\le \\pi$.\nIts [[Fourier series|G12KXDHPPS]] is $2 \\sum_{n \\ge 1} \\frac{(-1)^{n+1}}{n} \\sin nx$: indeed $f$ is odd, so the cosine coefficients vanish, and integration by parts gives\n$$b_n = \\frac{2}{\\pi} \\int_0^\\pi x \\sin nx \\, dx = \\frac{2(-1)^{n+1}}{n}.$$\nIn complex form, $\\hat f(0) = 0$ and $\\abs{\\hat f(n)} = \\frac{1}{\\abs{n}}$ for $n \\ne 0$.\nParseval's identity, from the [[L2 convergence of Fourier series|3DGHC142FT]], gives\n$$\\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} x^2 \\, dx = \\sum_{n \\ne 0} \\frac{1}{n^2}, \\qquad \\text{that is} \\qquad \\frac{\\pi^2}{3} = 2 \\sum_{n=1}^{\\infty} \\frac{1}{n^2}.$$\nSo $\\sum_{n=1}^{\\infty} \\frac{1}{n^2} = \\frac{\\pi^2}{6}$, which is the [[Basel problem|0DMY75D6FV]].\n\n## Remarks\n\nA second route uses the pointwise convergence of a Fourier series instead of Parseval.\nThe $2\\pi$-periodic function equal to $x^2$ on $[-\\pi, \\pi]$ is continuous and piecewise smooth, and its Fourier series $\\frac{\\pi^2}{3} + 4 \\sum_{n \\ge 1} \\frac{(-1)^n}{n^2} \\cos nx$ converges to it everywhere.\nAt $x = \\pi$ this gives $\\pi^2 = \\frac{\\pi^2}{3} + 4 \\sum_{n \\ge 1} \\frac{1}{n^2}$, the same value.\nParseval applied to $x^2$ gives $\\sum_{n \\ge 1} \\frac{1}{n^4} = \\frac{\\pi^4}{90}$ in the same way.\n\n## Explanations\n\n### L1 human\n\nThe energy of the function $x$ on one period can be computed in two ways: directly, or as the sum of the energies of its waves.\nThe waves of $x$ have amplitudes $1/n$, so their energies are $1/n^2$.\nComparing the two answers forces the sum $1 + \\frac{1}{4} + \\frac{1}{9} + \\cdots$ to equal $\\frac{\\pi^2}{6}$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "illustrates": [
      "3DGHC142FT"
    ],
    "kind": "example",
    "msc": [
      "42A16",
      "40A05"
    ],
    "name": "Basel problem via Fourier series",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Basel_problem"
      }
    ],
    "tag": "TYJWZDK485",
    "uses": [
      "0DMY75D6FV",
      "G12KXDHPPS",
      "3DGHC142FT"
    ]
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/TYJWZDK485/",
  "links": [
    {
      "field": "illustrates",
      "target": "3DGHC142FT"
    },
    {
      "field": "uses",
      "target": "0DMY75D6FV"
    },
    {
      "field": "uses",
      "target": "G12KXDHPPS"
    },
    {
      "field": "uses",
      "target": "3DGHC142FT"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/TYJWZDK485.md",
  "msc": [
    "42A16",
    "40A05"
  ],
  "name": "Basel problem via Fourier series",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TYJ/TYJWZDK485.md",
  "status": null,
  "tag": "TYJWZDK485",
  "uri": "m2://TYJWZDK485",
  "url": "https://tamnd.github.io/math2/wiki/basel-problem-via-fourier-series/",
  "writing": {
    "tag": "TYJWZDK485",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}