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  "body": "\n## Statement\n\nFor every real $x$,\n$$e^x = \\sum_{n=0}^{\\infty} \\frac{x^n}{n!}.$$\nThis series is the [[Maclaurin series|043SJ5RF59]] of the exponential function. Its partial sums converge to $e^x$ uniformly on each bounded interval.\n\n## Remarks\n\nBy the [[derivative rule|0EX9GFJ0B3]], every derivative of $\\exp$ at $0$ equals $\\exp(0) = 1$. So the Maclaurin coefficients are $1/n!$.\nWith the series as the definition of $\\exp$, the first claim holds by definition. Uniform convergence follows from [[the result on power series|0BWXV7GADW]].\nThe partial sums give good approximations near $0$. For example, $1 + x + x^2/2$ approximates $e^x$ with error $O(x^3)$.\n\n## Explanations\n\n### L1 human\n\nThe exponential function equals the sum of its Taylor polynomials at $0$.\nAdding more terms gives better approximations of $e^x$, for every $x$.\n",
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