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  "body": "\n## Definition\n\nThe exponential function $\\exp \\colon \\mathbb{R} \\to \\mathbb{R}$ is defined by\n$$\\exp(x) = \\sum_{n=0}^{\\infty} \\frac{x^n}{n!} = 1 + x + \\frac{x^2}{2} + \\frac{x^3}{6} + \\cdots.$$\n\n## Remarks\n\nThe [[power series|0VP56B4EMX]] converges absolutely for every real $x$, because [[its radius of convergence is infinite|0PVJRACB9H]].\nOne also writes $e^x$ for $\\exp(x)$, where $e = \\exp(1)$ is the number [[e|0MVCFMW7S4]].\nOther common definitions use the limit $\\lim_{n \\to \\infty} (1 + x/n)^n$, or the equation $f' = f$ with $f(0) = 1$. They give the same function.\nMathlib defines `Real.exp` as the real part of the complex exponential series.\n\n## Explanations\n\n### L1 human\n\nThe exponential function is an infinite polynomial with the coefficients $1/n!$.\nIt grows faster than every polynomial, and its derivative is itself.\n",
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