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  "body": "\n## Statement\n\nLet $(a_n)$ be a sequence of positive real numbers, and put\n$$R_n = n \\left( \\frac{a_n}{a_{n+1}} - 1 \\right).$$\n\n1. If there are $r \u003e 1$ and $N$ with $R_n \\ge r$ for all $n \\ge N$, then $\\sum a_n$ [[converges|0CVVKY7GY2]].\n2. If there is $N$ with $R_n \\le 1$ for all $n \\ge N$, then $\\sum a_n$ [[diverges|0NBRP6093V]].\n\nIn particular, if $R_n \\to R$, the series converges when $R \u003e 1$ and diverges when $R \u003c 1$.\n\n## Remarks\n\nThis is [[Kummer's test|9JQEWK5405]] with $c_n = n$.\nThen $\\rho_n = n a_n / a_{n+1} - (n + 1) = R_n - 1$, and $\\sum 1/n$ diverges.\nFor $R = 1$ the limit form gives no answer, and [[Bertrand's test|6T3E3AXZT7]] can then help.\nExample: let $a_1 = 1/2$ and $a_{n+1} = a_n (2n+1)/(2n+2)$, so $a_n = \\frac{1 \\cdot 3 \\cdots (2n-1)}{2 \\cdot 4 \\cdots (2n)}$.\nThe [[ratio test|0QNY6ZBXPN]] fails, since $a_{n+1}/a_n \\to 1$.\nHere $R_n = n/(2n+1) \\le 1$, so $\\sum a_n$ diverges.\nFor $a_n = 1/n^p$ one gets $R_n \\to p$, which recovers the [[p-series test|0ZNSSK8ZK1]] for $p \\ne 1$.\n\n## Explanations\n\n### L1 human\n\nRaabe's test is a sharper version of the ratio test.\nIt is used when the ratio of consecutive terms tends to $1$.\nIt measures how fast the ratio approaches $1$: a gap like $p/n$ with $p \u003e 1$ means convergence.\n\n### L2 human\n\nThe ratio test compares a series with a geometric series.\nRaabe's test compares it with the $p$-series $\\sum 1/n^p$ instead, through Kummer's test with weights $c_n = n$.\nIt settles many series whose terms are quotients of products, such as binomial and hypergeometric coefficients.\n[[Gauss's test|A7MJXJERNY]] is a convenient form when $a_n / a_{n+1}$ has an expansion in powers of $1/n$.\n",
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