{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nWe check the three claims in turn: the identity, the Maclaurin coefficients and the uniform convergence.\n\n1. By the definition of the [[exponential function|0RMVS7RXZX]], $e^x = \\exp(x) = \\sum_{n=0}^{\\infty} \\frac{x^n}{n!}$ for every real $x$. The series converges for every $x$, since its [[radius of convergence is infinite|0PVJRACB9H]].\n2. By [[the derivative of exp|0EX9GFJ0B3]], $\\exp$ is differentiable on $\\R$ and $\\exp' = \\exp$.\n3. We show by induction on $n$ that the [[higher derivative|084NGSTWM7]] $\\exp^{(n)}$ exists on $\\R$ and equals $\\exp$. For $n = 0$ this is the definition $\\exp^{(0)} = \\exp$.\n4. If $\\exp^{(n)} = \\exp$, then $\\exp^{(n)}$ is differentiable by step 2, and $\\exp^{(n+1)} = \\exp' = \\exp$. This completes the induction.\n5. Each $\\exp^{(n)} = \\exp$ is differentiable, hence [[continuous|07ZMX4ANJS]]. So $\\exp$ is of [[class $C^k$|06SKCHDQT5]] on $\\R$ for each $k$, that is, [[smooth|0BE3QDH73A]].\n6. At $0$, every term of the series with $n \\ge 1$ is $0$, and the term with $n = 0$ is $1$. So $\\exp^{(n)}(0) = \\exp(0) = 1$ for every $n$.\n7. By step 6, the [[Maclaurin series|043SJ5RF59]] of $\\exp$ is $\\sum_{n=0}^{\\infty} \\frac{\\exp^{(n)}(0)}{n!} x^n = \\sum_{n=0}^{\\infty} \\frac{x^n}{n!}$. This is the given series.\n8. Let $J$ be a bounded interval. Then $J \\subseteq [-r, r]$ for some real $r \u003e 0$.\n9. The radius of convergence is $\\infty \u003e r$. By the result on [[uniform convergence inside the radius|0BWXV7GADW]], the partial sums converge [[uniformly|08FPABDQ4C]] on $[-r, r]$ to $\\exp$.\n10. For each $\\varepsilon \u003e 0$, the index $N$ given by step 9 for $[-r, r]$ also works for every point of $J$. So the partial sums converge to $\\exp$ uniformly on $J$.\n\n## Idea\n\nThe exponential function is defined by this series, so the identity holds by definition. Because $\\exp$ is its own derivative, all its derivatives at $0$ equal $1$, which gives the coefficients $1/n!$. An infinite radius of convergence gives uniform convergence on every bounded interval.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "33B10"
    ],
    "name": "Proof of the power series of the exponential function",
    "proves": "06JZGMYTYD",
    "sources": [
      {
        "locator": "Power series",
        "url": "https://en.wikipedia.org/wiki/Exponential_function"
      }
    ],
    "tag": "T4HQMF76W8",
    "uses": [
      "0RMVS7RXZX",
      "0PVJRACB9H",
      "0EX9GFJ0B3",
      "07ZMX4ANJS",
      "084NGSTWM7",
      "06SKCHDQT5",
      "0BE3QDH73A",
      "043SJ5RF59",
      "0BWXV7GADW",
      "08FPABDQ4C"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/T4HQMF76W8/",
  "links": [
    {
      "field": "proves",
      "target": "06JZGMYTYD"
    },
    {
      "field": "uses",
      "target": "0RMVS7RXZX"
    },
    {
      "field": "uses",
      "target": "0PVJRACB9H"
    },
    {
      "field": "uses",
      "target": "0EX9GFJ0B3"
    },
    {
      "field": "uses",
      "target": "07ZMX4ANJS"
    },
    {
      "field": "uses",
      "target": "084NGSTWM7"
    },
    {
      "field": "uses",
      "target": "06SKCHDQT5"
    },
    {
      "field": "uses",
      "target": "0BE3QDH73A"
    },
    {
      "field": "uses",
      "target": "043SJ5RF59"
    },
    {
      "field": "uses",
      "target": "0BWXV7GADW"
    },
    {
      "field": "uses",
      "target": "08FPABDQ4C"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/T4HQMF76W8.md",
  "msc": [
    "33B10"
  ],
  "name": "Proof of the power series of the exponential function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/T4H/T4HQMF76W8.md",
  "status": null,
  "tag": "T4HQMF76W8",
  "uri": "m2://T4HQMF76W8",
  "url": "https://math2.tamnd.com/wiki/proof-of-the-power-series-of-the-exponential-function/",
  "writing": {
    "tag": "T4HQMF76W8",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}