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  "body": "\n## Statement\n\nLet $(a_n)$ and $(c_n)$ be sequences of positive real numbers.\nFor each $n$ put\n$$\\rho_n = c_n \\frac{a_n}{a_{n+1}} - c_{n+1}.$$\n\n1. If there are $\\delta \u003e 0$ and $N$ with $\\rho_n \\ge \\delta$ for all $n \\ge N$, then $\\sum a_n$ [[converges|0CVVKY7GY2]].\n2. If $\\sum 1/c_n$ diverges and there is $N$ with $\\rho_n \\le 0$ for all $n \\ge N$, then $\\sum a_n$ [[diverges|0NBRP6093V]].\n\n## Remarks\n\nFor part 1, multiply $\\rho_n \\ge \\delta$ by $a_{n+1}$ to get $c_n a_n - c_{n+1} a_{n+1} \\ge \\delta a_{n+1}$ for $n \\ge N$.\nSumming from $N$ to $m$ telescopes, so $\\delta \\sum_{n=N+1}^{m+1} a_n \\le c_N a_N$, and the partial sums are bounded.\nFor part 2, the sequence $c_n a_n$ increases from $n = N$ on, so $a_n \\ge c_N a_N / c_n$, and the [[comparison test|04Z5G0THN6]] with $\\sum 1/c_n$ gives divergence.\nThe choice $c_n = 1$ gives the [[ratio test|0QNY6ZBXPN]], the choice $c_n = n$ gives [[Raabe's test|CD6CAQ6XDN]], and the choice $c_n = n \\ln n$ gives [[Bertrand's test|6T3E3AXZT7]].\n\n## Explanations\n\n### L1 human\n\nKummer's test is a family of tests, one for each choice of comparison weights $c_n$.\nIt compares the decay of the terms $a_n$ with the growth of the weights.\nThe familiar ratio test is the simplest member of the family.\n\n### L2 human\n\nWhen the ratio test gives the limit $1$, one uses Kummer's test with weights that grow, such as $n$ or $n \\ln n$.\nEach such choice gives a finer test: Raabe's test catches the $p$-series, and Bertrand's test catches series like $\\sum 1/(n (\\ln n)^p)$.\nThe proof only uses telescoping sums and the comparison test.\n",
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