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  "body": "\n## Statement\n\nLet $(a_n)$ be a sequence of positive real numbers.\nAssume that there are a real number $h$, a real number $s \u003e 1$ and a bounded real sequence $(\\beta_n)$ with\n$$\\frac{a_n}{a_{n+1}} = 1 + \\frac{h}{n} + \\frac{\\beta_n}{n^s}$$\nfor all $n \\ge 1$.\nThen $\\sum a_n$ [[converges|0CVVKY7GY2]] if $h \u003e 1$, and [[diverges|0NBRP6093V]] if $h \\le 1$.\n\n## Remarks\n\nThe quantity of [[Raabe's test|CD6CAQ6XDN]] is $n(a_n/a_{n+1} - 1) = h + \\beta_n n^{1-s}$, which tends to $h$.\nSo Raabe's test settles the cases $h \u003e 1$ and $h \u003c 1$.\nFor $h = 1$, the quantity of [[Bertrand's test|6T3E3AXZT7]] is $\\beta_n n^{1-s} \\ln n$, which tends to $0$, so the series diverges.\nExample: let $a, b, c \u003e 0$ and let $a_n$ be the $n$-th term of the hypergeometric series $\\sum_n \\frac{(a)_n (b)_n}{(c)_n \\, n!}$, where $(x)_n = x(x+1)\\cdots(x+n-1)$.\nThen $a_n / a_{n+1} = \\frac{(n + c)(n + 1)}{(n + a)(n + b)} = 1 + \\frac{c + 1 - a - b}{n} + O(1/n^2)$.\nSo the series converges exactly when $c \u003e a + b$.\n\n## Explanations\n\n### L1 human\n\nGauss's test reads off convergence from the expansion of the ratio $a_n / a_{n+1}$ in powers of $1/n$.\nOnly the coefficient $h$ of $1/n$ matters: above $1$ the series converges, otherwise it diverges.\nIt is the natural test for terms that are products of linear factors in $n$.\n\n### L2 human\n\nGauss's test packages Raabe's test and Bertrand's test into one statement for ratios with an asymptotic expansion.\nA classical application is the convergence of the hypergeometric series at $x = 1$, as in the example above.\nThe error term must be $O(1/n^s)$ with $s \u003e 1$; an error like $1/(n \\ln n)$ would hide the boundary case.\n",
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