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  "body": "\n## Statement\n\nLet $e = \\exp(1)$, where $\\exp$ is the [[exponential function|0RMVS7RXZX]].\nThen\n$$\\lim_{n \\to \\infty} \\left(1 + \\frac{1}{n}\\right)^n = e.$$\nMore generally, for each real number $x$,\n$$\\lim_{n \\to \\infty} \\left(1 + \\frac{x}{n}\\right)^n = e^x.$$\n\n## Remarks\n\nSome texts take the limit as the definition of [[the number e|0MVCFMW7S4]]. The sequence $(1 + 1/n)^n$ is increasing and bounded above by $3$, so the [[monotone convergence theorem|0FM6N4R0VB]] gives the limit.\nThe first values are $2$, $2.25$, about $2.370$ and about $2.441$. The limit is $e \\approx 2.71828$.\nJacob Bernoulli found this limit when he studied compound interest.\nMathlib states the general form as `Real.tendsto_one_add_div_pow_exp`.\n\n## Explanations\n\n### L1 human\n\nPut one unit of money in a bank at $100\\%$ yearly interest, paid $n$ times a year.\nAfter one year you have $(1 + 1/n)^n$ units. As $n$ grows, this approaches $e$, and it never exceeds $e$.\n",
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