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  "body": "\n## Definition\n\nLet $P$ be a property of real or complex sequences $(a_n)$.\nThe property $P$ is a convergence test if every [[series|0JH4MN4K1C]] $\\sum a_n$ whose terms have $P$ is [[convergent|0CVVKY7GY2]].\nIt is a divergence test if every series $\\sum a_n$ whose terms have $P$ is [[divergent|0NBRP6093V]].\n\n## Remarks\n\nExample: the property \"there are $q \u003c 1$ and $N$ with $\\abs{a_{n+1}} \\le q \\abs{a_n}$ for all $n \\ge N$\" is a convergence test.\nIndeed $\\abs{a_n} \\le \\abs{a_N} q^{n - N}$ for $n \\ge N$, and the [[comparison test|04Z5G0THN6]] with a [[geometric series|01X5CVV1Q5]] gives convergence.\nNon-example: the property \"$a_n \\to 0$\" is not a convergence test.\nThe terms $1/n$ of the [[harmonic series|0CXTMAY5PX]] tend to $0$, but the series diverges.\nIts negation \"$a_n \\not\\to 0$\" is a divergence test.\nA test is inconclusive on a series when the series has neither the property of the test nor the property of its companion divergence test.\n\n## Explanations\n\n### L1 human\n\nA convergence test is a sufficient condition: when it holds, convergence is certain.\nWhen it does not hold, nothing follows unless another test applies.\n",
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