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  "body": "\n## Statement\n\nLet $(a_n)$ be a real sequence with $a_n \\ne 0$ for all $n$.\nAssume that $|a_{n+1}| / |a_n| \\to L$ as $n \\to \\infty$, for a real number $L$ with $L \u003e 1$.\nThen the series $\\sum a_n$ diverges.\n\n## Remarks\n\nFor large $n$ we have $|a_{n+1}| \u003e |a_n|$. So $|a_n|$ is eventually increasing and positive, and $a_n$ does not tend to $0$. The [[term test|0VVY4H92G2]] gives divergence.\nThe same proof works when $|a_{n+1}| / |a_n| \\to +\\infty$.\nFor example, $\\sum 2^n / n$ diverges, since the ratio is $2n/(n+1) \\to 2$.\n\n## Explanations\n\n### L1 human\n\nIf each term is eventually bigger than the term before, the terms cannot shrink to $0$.\nThen the series cannot converge.\n",
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