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  "body": "\n## Example\n\nLet $r$ be a real number with $r \u003e 1$.\nThen $r^n \\to +\\infty$ as $n \\to \\infty$, in the sense of a [[sequence tending to infinity|09AZCV6GMC]].\nWrite $r = 1 + h$ with $h \u003e 0$. By Bernoulli's inequality, $r^n \\ge 1 + n h$ for each natural number $n$.\nLet $M$ be a real number, and choose a natural number $N$ with $N h \u003e M$. Then $r^n \u003e M$ for all $n \\ge N$.\n\n## Remarks\n\nFor $r \u003c -1$ the sequence $r^n$ does not tend to $+\\infty$ or to $-\\infty$. Its terms change sign, but $|r^n| \\to +\\infty$.\nExponential growth is faster than polynomial growth. The node [[exponentials dominate powers in sequences|0MVTCFW42K]] makes this precise.\n\n## Explanations\n\n### L1 human\n\nDoubling again and again gives $1, 2, 4, 8, 16, \\dots$, and these numbers grow without bound.\nEach factor $r \u003e 1$ does the same, but a factor close to $1$ needs more steps.\n",
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