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  "body": "\n## Statement\n\nLet $(a_n)_{n \\ge 2}$ be a sequence of positive real numbers, and put\n$$B_n = \\ln n \\left( n \\left( \\frac{a_n}{a_{n+1}} - 1 \\right) - 1 \\right),$$\nwhere $\\ln$ is the [[natural logarithm|0044CPHJBV]].\n\n1. If there are $r \u003e 1$ and $N$ with $B_n \\ge r$ for all $n \\ge N$, then $\\sum a_n$ [[converges|0CVVKY7GY2]].\n2. If there is $N$ with $B_n \\le 1$ for all $n \\ge N$, then $\\sum a_n$ [[diverges|0NBRP6093V]].\n\nIn particular, if $B_n \\to B$, the series converges when $B \u003e 1$ and diverges when $B \u003c 1$.\n\n## Remarks\n\nThis is [[Kummer's test|9JQEWK5405]] with $c_n = n \\ln n$.\nA short computation gives $\\rho_n = B_n - (n+1) \\ln(1 + 1/n)$.\nThe inequality $\\ln(1 + x) \\ge x/(1 + x)$ gives $(n+1)\\ln(1 + 1/n) \\ge 1$, and this quantity tends to $1$.\nSo $B_n \\ge r \u003e 1$ gives $\\rho_n \\ge (r - 1)/2$ for large $n$, and $B_n \\le 1$ gives $\\rho_n \\le 0$; moreover $\\sum 1/(n \\ln n)$ diverges.\nExample: if $a_n / a_{n+1} = 1 + 1/n + O(1/n^s)$ with $s \u003e 1$, then $B_n = \\ln n \\cdot O(n^{1-s}) \\to 0$, so the series diverges.\nThis is the case that [[Raabe's test|CD6CAQ6XDN]] leaves open in [[Gauss's test|A7MJXJERNY]].\nFor $a_n = 1/(n (\\ln n)^p)$ one finds $B_n \\to p$, so the test decides these series for $p \\ne 1$.\n\n## Explanations\n\n### L1 human\n\nBertrand's test is the next step after Raabe's test.\nIt is used when Raabe's quantity tends to $1$.\nIt compares the series with the slowly converging or diverging series $\\sum 1/(n (\\ln n)^p)$.\n\n### L2 human\n\nThe ratio test, Raabe's test and Bertrand's test form a scale of finer and finer tests, all special cases of Kummer's test.\nEach one looks at the next term in the expansion of $a_n / a_{n+1}$.\nOne use of Bertrand's test is the proof of Gauss's test in the boundary case $h = 1$.\n",
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