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  "body": "\n## Statement\n\nLet $\\sum a_n$ be a [[convergent|0CVVKY7GY2]] real series.\nLet $(b_n)$ be a [[monotone|0KN22AAW5R]] and [[bounded|05FQ3VGAR9]] real sequence.\nThen the series $\\sum a_n b_n$ converges.\n\n## Remarks\n\nThe sequence $(b_n)$ converges to some $b$, by the [[monotone convergence theorem|0FM6N4R0VB]]. Then $\\sum a_n b_n = \\sum a_n (b_n - b) + b \\sum a_n$.\nThe first series converges by [[Dirichlet's test|0V9ME8GQE7]], since $b_n - b$ is monotone with limit $0$.\nThe hypotheses on $\\sum a_n$ are stronger than in Dirichlet's test, and those on $(b_n)$ are weaker.\n\n## History\n\nNiels Henrik Abel proved the test in 1826.\n\n## Explanations\n\n### L1 human\n\nMultiplying the terms of a convergent series by a bounded monotone sequence of weights keeps convergence.\nThe weights settle down to a limit, so they cannot disturb the series too much.\n",
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