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  "body": "\n## Statement\n\nLet $\\sum a_n$ be a [[convergent series|0CVVKY7GY2]] of real numbers.\nThen $a_n \\to 0$ as $n \\to \\infty$.\nEquivalently, if $(a_n)$ does not converge to $0$, then the series $\\sum a_n$ diverges.\n\n## Remarks\n\nFor the proof, let $s$ be the sum. Then $a_n = s_n - s_{n-1} \\to s - s = 0$.\nThe converse is false. The terms of the [[harmonic series|0CXTMAY5PX]] tend to $0$, and the series diverges.\nFor example, $\\sum (-1)^n$ and $\\sum n/(n+1)$ diverge by this test.\n\n## Explanations\n\n### L1 human\n\nIf you keep adding terms that do not get small, the total cannot settle down.\nSo small terms are necessary for convergence, but they are not enough.\n",
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