{
  "aliases": [
    "d-alembert-ratio-test"
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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 $L \u003c 1$.\nThen the series $\\sum a_n$ converges [[absolutely|0W3K3QPEWT]].\n\n## Remarks\n\nChoose $q$ with $L \u003c q \u003c 1$. For $n \\ge N$ we get $|a_{n+1}| \\le q |a_n|$, so $|a_n| \\le |a_N| q^{n-N}$. The [[comparison test|04Z5G0THN6]] with a [[geometric series|01X5CVV1Q5]] gives the result.\nFor $L \u003e 1$ the series diverges, as the [[divergence part of the ratio test|08AM0KRCZ1]] states.\nFor $L = 1$ the test gives no answer. The series $\\sum 1/n$ diverges and $\\sum 1/n^2$ converges, and both have $L = 1$.\nFor example, $\\sum x^n / n!$ converges for each real $x$, since the ratio is $|x|/(n+1) \\to 0$.\n\n## Explanations\n\n### L1 human\n\nIf each term is eventually less than a fixed fraction $q \u003c 1$ of the term before, the terms shrink at least as fast as $q^n$.\nSo the series behaves at least as well as a convergent geometric series.\n",
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  "name": "Ratio test",
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