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  "body": "\n## Proof\n\n1. Apply the definition of the [[limit|0XKWSSY8P2]] with $\\varepsilon = L - 1 \u003e 0$. There is $N$ with $\\abs{a_{n+1}}/\\abs{a_n} \u003e L - \\varepsilon = 1$ for all $n \\ge N$.\n2. So $\\abs{a_{n+1}} \u003e \\abs{a_n}$ for $n \\ge N$. By induction, $\\abs{a_n} \\ge \\abs{a_N}$ for all $n \\ge N$.\n3. By hypothesis $a_N \\ne 0$, so $\\delta = \\abs{a_N} \u003e 0$.\n4. If $a_n \\to 0$, then $\\abs{a_n} \u003c \\delta$ for all large $n$. This contradicts step 2.\n5. So $(a_n)$ does not converge to $0$. By the [[term test|0VVY4H92G2]], the series $\\sum a_n$ diverges.\n\n## Idea\n\nWhen the ratio tends to a number above $1$, the absolute values of the terms eventually grow. Growing nonzero terms cannot tend to $0$, so the series fails the term test.\n",
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