{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "02YRQXZDCS"
    },
    {
      "field": "uses",
      "target": "0BJSWZJG18"
    },
    {
      "field": "uses",
      "target": "0CN97GADE5"
    },
    {
      "field": "uses",
      "target": "0K0H5E3Z5N"
    },
    {
      "field": "uses",
      "target": "0MX1HZBBXJ"
    },
    {
      "field": "uses",
      "target": "0NBRP6093V"
    },
    {
      "field": "uses",
      "target": "0S4EMDQS2F"
    },
    {
      "field": "uses",
      "target": "0VVY4H92G2"
    },
    {
      "field": "uses",
      "target": "0W3K3QPEWT"
    },
    {
      "field": "uses",
      "target": "0ZNSSK8ZK1"
    },
    {
      "field": "uses",
      "target": "2DJT2AYY23"
    },
    {
      "field": "uses",
      "target": "4670W2WNFX"
    },
    {
      "field": "uses",
      "target": "5VN386XHGA"
    },
    {
      "field": "uses",
      "target": "6T3E3AXZT7"
    },
    {
      "field": "uses",
      "target": "8AX4XREEPX"
    },
    {
      "field": "uses",
      "target": "8SK3GQ4S0M"
    },
    {
      "field": "uses",
      "target": "9JQEWK5405"
    },
    {
      "field": "uses",
      "target": "A7MJXJERNY"
    },
    {
      "field": "uses",
      "target": "AEZNCY3N3K"
    },
    {
      "field": "uses",
      "target": "B54HPY6C5Z"
    },
    {
      "field": "uses",
      "target": "BWWZB1F0HY"
    },
    {
      "field": "uses",
      "target": "C28YT5DQFF"
    },
    {
      "field": "uses",
      "target": "CD6CAQ6XDN"
    },
    {
      "field": "uses",
      "target": "GA7Q813F50"
    },
    {
      "field": "uses",
      "target": "HNCY0067F1"
    },
    {
      "field": "uses",
      "target": "MA9EMC488R"
    },
    {
      "field": "uses",
      "target": "RZKC8B00HZ"
    },
    {
      "field": "uses",
      "target": "S7A8YFGG8X"
    },
    {
      "field": "uses",
      "target": "X629NF2CQS"
    },
    {
      "field": "uses",
      "target": "Y0JH94QR80"
    },
    {
      "field": "uses",
      "target": "Z35TD4ZZRD"
    },
    {
      "field": "about",
      "target": "B54HPY6C5Z"
    }
  ],
  "body": "\n## Definition\n\nLet $\\sum a_n$ be a [[series|0JH4MN4K1C]] of real numbers with [[partial sums|0NB643K60Q]] $s_N = \\sum_{n=0}^{N} a_n$.\nThe series is convergent if the sequence $(s_N)$ is a [[convergent sequence|0AZ3R52593]].\n\n## Remarks\n\nThe series $\\sum 1/2^n$ converges, since its partial sums $2 - 1/2^N$ tend to $2$.\nThe limit of the partial sums is the [[sum of the series|0K0H5E3Z5N]].\nA series that does not converge is a [[divergent series|0NBRP6093V]].\nChanging finitely many terms does not change whether a series converges. It can change the sum.\n\n## Explanations\n\n### L1 human\n\nA series converges when its running totals approach a fixed number.\nAdding more and more terms then changes the total less and less.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "0AF7X08PCR",
    "inventory": [
      "ANWN0Z1HYT"
    ],
    "kind": "definition",
    "msc": [
      "40A05"
    ],
    "name": "Convergent series",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Convergent_series"
      }
    ],
    "tag": "0CVVKY7GY2",
    "uses": [
      "0JH4MN4K1C",
      "0NB643K60Q",
      "0AZ3R52593"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/0CVVKY7GY2/",
  "links": [
    {
      "field": "defines",
      "target": "0AF7X08PCR"
    },
    {
      "field": "uses",
      "target": "0JH4MN4K1C"
    },
    {
      "field": "uses",
      "target": "0NB643K60Q"
    },
    {
      "field": "uses",
      "target": "0AZ3R52593"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/0CVVKY7GY2.md",
  "msc": [
    "40A05"
  ],
  "name": "Convergent series",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0CV/0CVVKY7GY2.md",
  "status": null,
  "tag": "0CVVKY7GY2",
  "uri": "m2://0CVVKY7GY2",
  "url": "https://tamnd.github.io/math2/wiki/convergent-series-definition/",
  "writing": {
    "tag": "0CVVKY7GY2",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}