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  "body": "\n## Proof\n\nWe use the ratio formula for the radius of convergence.\n\n1. The series $\\sum_{n=0}^{\\infty} \\frac{x^n}{n!}$ is a [[power series|0VP56B4EMX]] with center $0$ and coefficients $a_n = \\frac{1}{n!}$. Each $a_n$ is nonzero.\n2. By the definition of the [[factorial|0WFCGVMRRC]], $(n+1)! = (n+1) \\cdot n!$. So $\\frac{\\abs{a_{n+1}}}{\\abs{a_n}} = \\frac{n!}{(n+1)!} = \\frac{1}{n+1}$.\n3. We show $\\frac{1}{n+1} \\to 0$. Let $\\varepsilon \u003e 0$. By the [[Archimedean property|0A74SX55QP]], there is a natural number $N$ with $N \u003e 1/\\varepsilon$. For $n \\ge N$, $0 \u003c \\frac{1}{n+1} \u003c \\frac{1}{N} \u003c \\varepsilon$.\n4. By steps 2 and 3, the hypotheses of the [[ratio formula|0JBMYAA9QH]] hold with $L = 0$. So the [[radius of convergence|0660QX1VPR]] is $R = 1/0 = \\infty$.\n5. Let $x$ be real, and let $r = \\abs{x} + 1$. Then $0 \u003c r \u003c R$. By the result on [[uniform convergence inside the radius|0BWXV7GADW]], the series converges absolutely on $[-r, r]$.\n6. Since $x \\in [-r, r]$, the series [[converges absolutely|0W3K3QPEWT]] at $x$. This holds for every real $x$.\n\n## Idea\n\nEach coefficient is the previous one divided by $n + 1$. So the coefficients shrink faster than any geometric sequence, and the ratio formula gives an infinite radius. Inside the radius a power series converges absolutely, and here the radius covers every real number.\n",
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