{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\n1. Let $s_N = \\sum_{n=0}^{N} a_n$ be the [[partial sums|0NB643K60Q]]. Since the series is a [[convergent series|0CVVKY7GY2]], $s_N \\to S$ for a real number $S$, the [[sum of the series|0K0H5E3Z5N]].\n2. For each $n \\ge 1$, the partial sums satisfy $a_n = s_n - s_{n-1}$.\n3. Let $\\varepsilon \u003e 0$. By the definition of the [[limit|0XKWSSY8P2]], there is $N$ with $\\abs{s_m - S} \u003c \\varepsilon / 2$ for all $m \\ge N$.\n4. Let $n \\ge N + 1$. Then both $n \\ge N$ and $n - 1 \\ge N$.\n5. By step 2 and the [[triangle inequality|03HF5PMY5T]], $\\abs{a_n} = \\abs{(s_n - S) + (S - s_{n-1})} \\le \\abs{s_n - S} + \\abs{s_{n-1} - S}$.\n6. By steps 3 and 4, the right side of step 5 is less than $\\varepsilon/2 + \\varepsilon/2 = \\varepsilon$.\n7. So $\\abs{a_n - 0} \u003c \\varepsilon$ for all $n \\ge N + 1$. Since $\\varepsilon \u003e 0$ was arbitrary, $a_n \\to 0$.\n8. The second form is the contrapositive of the first. If $(a_n)$ does not converge to $0$, the series cannot converge, by steps 1 to 7. So it diverges.\n\n## Idea\n\nEach term is the difference of two consecutive partial sums. When the partial sums settle down to a limit, consecutive ones get close to each other, so their difference tends to $0$.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "40A05"
    ],
    "name": "Proof of the term test",
    "proves": "0VVY4H92G2",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Term_test"
      }
    ],
    "tag": "SZ5WPYWAHQ",
    "uses": [
      "0CVVKY7GY2",
      "0NB643K60Q",
      "0K0H5E3Z5N",
      "0XKWSSY8P2",
      "03HF5PMY5T"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/SZ5WPYWAHQ/",
  "links": [
    {
      "field": "proves",
      "target": "0VVY4H92G2"
    },
    {
      "field": "uses",
      "target": "0CVVKY7GY2"
    },
    {
      "field": "uses",
      "target": "0NB643K60Q"
    },
    {
      "field": "uses",
      "target": "0K0H5E3Z5N"
    },
    {
      "field": "uses",
      "target": "0XKWSSY8P2"
    },
    {
      "field": "uses",
      "target": "03HF5PMY5T"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/SZ5WPYWAHQ.md",
  "msc": [
    "40A05"
  ],
  "name": "Proof of the term test",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SZ5/SZ5WPYWAHQ.md",
  "status": null,
  "tag": "SZ5WPYWAHQ",
  "uri": "m2://SZ5WPYWAHQ",
  "url": "https://math2.tamnd.com/wiki/proof-of-the-term-test/",
  "writing": {
    "tag": "SZ5WPYWAHQ",
    "kind": "proof",
    "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
      }
    ]
  }
}