{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $G$ be an additive abelian group, let $F \\colon \\Z \\to G$ be a function, and let $\\Delta F(k) = F(k + 1) - F(k)$ be its [[forward difference|TXTR68V4R1]].\nThen for all integers $a \\le b$,\n$$\\sum_{k=a}^{b-1} \\Delta F(k) = F(b) - F(a).$$\nEquivalently, if $F$ is an [[indefinite sum|5VK9NZM0R7]] of $f \\colon \\Z \\to G$, then $\\sum_{k=a}^{b-1} f(k) = F(b) - F(a)$.\n\n## Remarks\n\nThe proof is a telescoping sum: $\\sum_{k=a}^{b-1} (F(k+1) - F(k))$ cancels to $F(b) - F(a)$, as in a [[telescoping series|0GXKM1R7E4]].\nFor $a = b$ both sides are $0$.\nExample: $\\Delta (k)_3 = 3 (k)_2$ for the falling factorial $(k)_3 = k(k-1)(k-2)$, so $\\sum_{k=0}^{n-1} k(k-1) = \\frac{n(n-1)(n-2)}{3}$.\nA related identity, with $\\Delta$ of step $h$ on any additive commutative monoid, is $\\sum_{k=0}^{n-1} f(x + kh) = \\sum_{j=0}^{n-1} \\binom{n}{j+1} \\Delta_h^j f(x)$, obtained by summing [[Newton's forward difference formula|DDHCMKHD2T]].\n\n## Explanations\n\n### L1 human\n\nAdding up all the steps of a function gives its total change.\nThis is the discrete version of $\\int_a^b F'(x)\\,dx = F(b) - F(a)$.\nTo sum $f(a) + \\dots + f(b-1)$, it is enough to find one function $F$ whose steps are the values of $f$.\n\n### L2 human\n\nThe theorem reduces the evaluation of finite sums to finding indefinite sums, just as the [[fundamental theorem of calculus|0NHGY0N0JB]] reduces integrals to antiderivatives.\nA table of differences, such as $\\Delta (k)_m = m (k)_{m-1}$ for [[falling factorials|0SJNCJCCQG]] and $\\Delta c^k = (c - 1) c^k$, becomes a table of sums.\nTo sum $k^2$, write $k^2 = (k)_2 + (k)_1$, which gives $\\sum_{k=0}^{n-1} k^2 = \\frac{(n)_3}{3} + \\frac{(n)_2}{2}$.\n[[Summation by parts|9G82Z17DKA]] is the discrete analogue of integration by parts and is proved from the same telescoping.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "SMAQN2N739",
      "FT9JQA58KJ"
    ],
    "kind": "statement",
    "msc": [
      "39A12",
      "39A70"
    ],
    "name": "Fundamental theorem of the calculus of finite differences",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Indefinite_sum"
      }
    ],
    "tag": "MNSSV4QHD1",
    "uses": [
      "TXTR68V4R1",
      "5VK9NZM0R7"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/MNSSV4QHD1/",
  "links": [
    {
      "field": "about",
      "target": "SMAQN2N739"
    },
    {
      "field": "about",
      "target": "FT9JQA58KJ"
    },
    {
      "field": "uses",
      "target": "TXTR68V4R1"
    },
    {
      "field": "uses",
      "target": "5VK9NZM0R7"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/MNSSV4QHD1.md",
  "msc": [
    "39A12",
    "39A70"
  ],
  "name": "Fundamental theorem of the calculus of finite differences",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MNS/MNSSV4QHD1.md",
  "status": null,
  "tag": "MNSSV4QHD1",
  "uri": "m2://MNSSV4QHD1",
  "url": "https://tamnd.github.io/math2/wiki/fundamental-theorem-of-the-calculus-of-finite-differences/",
  "writing": {
    "tag": "MNSSV4QHD1",
    "kind": "statement",
    "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
      }
    ]
  }
}