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  "body": "\n## Statement\n\nLet $k$ be a natural number and let $r$ be a real number with $r \u003e 1$.\nThen\n$$\\lim_{n \\to \\infty} \\frac{n^k}{r^n} = 0.$$\n\n## Remarks\n\nFor $k = 1$ and $r = 2$, it says $n / 2^n \\to 0$. The terms are $1/2, 2/4, 3/8, 4/16, \\dots$.\nOne proof uses the ratio of consecutive terms. The ratio $\\frac{(n+1)^k}{n^k} \\cdot \\frac{1}{r}$ tends to $1/r \u003c 1$, so the terms eventually shrink by a fixed factor.\nIn the language of [[little o notation|04TSHB9H8D]], it says $n^k = o(r^n)$.\nThe node [[exponential growth beats polynomial growth|0CT6CBGTYY]] gives the version for real variables.\n\n## Explanations\n\n### L1 human\n\nA power such as $n^{10}$ grows fast, but an exponential such as $1.01^n$ grows faster in the end.\nFor small $n$ the power can be much larger. For large $n$ the exponential always wins.\n",
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