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  "body": "\n## Statement\n\nLet $(b_n)$ be a decreasing real sequence with $b_n \\to 0$ as $n \\to \\infty$.\nThen the [[alternating series|017BR1M98J]] $\\sum_{n=0}^{\\infty} (-1)^n b_n$ converges.\nMoreover, its sum $S$ and its partial sums $s_N = \\sum_{n=0}^{N} (-1)^n b_n$ satisfy $|S - s_N| \\le b_{N+1}$ for each $N$.\n\n## Remarks\n\nThe even partial sums $s_0, s_2, s_4, \\dots$ decrease, and the odd partial sums $s_1, s_3, \\dots$ increase. Each odd partial sum is below each even one, so both sequences converge by the [[monotone convergence theorem|0FM6N4R0VB]]. Their limits agree, since $s_{2k} - s_{2k+1} = b_{2k+1} \\to 0$.\nThe hypothesis that $(b_n)$ decreases is needed. Without it, an alternating series with terms tending to $0$ can diverge.\nMathlib states the convergence for an antitone sequence, in `Antitone.tendsto_alternating_series_of_tendsto_zero`.\n\n## History\n\nLeibniz discussed the criterion in an unpublished work of 1676. For this reason it is also called the Leibniz test.\n\n## Explanations\n\n### L1 human\n\nEach step goes in the other direction and is shorter than the step before.\nSo the partial sums swing back and forth with smaller and smaller swings, and they settle down.\n",
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  "name": "Alternating series test",
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