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  "body": "\n## Statement\n\nLet $(a_n)$ be a real sequence.\nThe series $\\sum a_n$ converges if and only if the following holds for each $\\varepsilon \u003e 0$.\nThere is a natural number $N$ such that\n$$\\left| \\sum_{k=m+1}^{n} a_k \\right| \u003c \\varepsilon \\quad \\text{for all natural numbers } n \u003e m \\ge N.$$\n\n## Remarks\n\nThe sum in the condition is $s_n - s_m$, a difference of [[partial sums|0NB643K60Q]]. So the condition says that $(s_n)$ is a Cauchy sequence, and the result is the [[Cauchy criterion for sequences|0ZA4AX4MW3]].\nWith $n = m + 1$ it gives the [[term test|0VVY4H92G2]].\nIt proves that [[absolutely convergent series converge|0Y123X48GA]].\n\n## Explanations\n\n### L1 human\n\nA series converges exactly when its tails become small.\nEach block of consecutive terms far out in the series must have a small sum.\n",
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