{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nThe alternating harmonic series is\n$$\\sum_{n=1}^{\\infty} \\frac{(-1)^{n+1}}{n} = 1 - \\frac{1}{2} + \\frac{1}{3} - \\frac{1}{4} + \\cdots.$$\nIt converges by the [[alternating series test|0PK8QH1W1V]], since $1/n$ decreases to $0$.\nIts sum is $\\ln 2 \\approx 0.6931$.\n\n## Remarks\n\nThe series of absolute values is the [[harmonic series|0CXTMAY5PX]], which diverges. So the convergence is [[conditional|02YRQXZDCS]].\nThe value $\\ln 2$ comes from the [[Mercator series|0YZXNE3HFR]] $\\ln(1 + x) = x - x^2/2 + x^3/3 - \\cdots$ at $x = 1$, with [[Abel's theorem|0S4EMDQS2F]].\nBy the [[Riemann rearrangement theorem|0E7VBYGFF7]], a reordering of the terms can give any real sum.\n\n## Explanations\n\n### L1 human\n\nThe partial sums are $1, 0.5, 0.833, 0.583, 0.783, \\dots$.\nThey swing around $\\ln 2 \\approx 0.693$ and close in on it slowly.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "instance_of": [
      "0AF7X08PCR",
      "0QED0GPYKY"
    ],
    "kind": "example",
    "msc": [
      "40A05"
    ],
    "name": "Alternating harmonic series",
    "sources": [
      {
        "locator": "Series representations",
        "url": "https://en.wikipedia.org/wiki/Natural_logarithm_of_2"
      },
      {
        "locator": "Alternating harmonic series",
        "url": "https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)"
      }
    ],
    "tag": "0JQ5Y3WWVE",
    "uses": [
      "0PK8QH1W1V",
      "0CXTMAY5PX"
    ]
  },
  "kind": "example",
  "link": "https://math2.tamnd.com/node/0JQ5Y3WWVE/",
  "links": [
    {
      "field": "instance_of",
      "target": "0AF7X08PCR"
    },
    {
      "field": "instance_of",
      "target": "0QED0GPYKY"
    },
    {
      "field": "uses",
      "target": "0PK8QH1W1V"
    },
    {
      "field": "uses",
      "target": "0CXTMAY5PX"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0JQ5Y3WWVE.md",
  "msc": [
    "40A05"
  ],
  "name": "Alternating harmonic series",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0JQ/0JQ5Y3WWVE.md",
  "status": null,
  "tag": "0JQ5Y3WWVE",
  "uri": "m2://0JQ5Y3WWVE",
  "url": "https://math2.tamnd.com/wiki/alternating-harmonic-series/",
  "writing": {
    "tag": "0JQ5Y3WWVE",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}