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  "body": "\n## Statement\n\nLet $(a_n)$ be a [[monotone|0KN22AAW5R]] real sequence with $a_n \\to 0$.\nLet $(b_n)$ be a real sequence whose [[partial sums|0NB643K60Q]] are bounded. That is, there is $M$ with $\\left| \\sum_{k=1}^{n} b_k \\right| \\le M$ for all $n$.\nThen the series $\\sum_{n=1}^{\\infty} a_n b_n$ converges.\n\n## Remarks\n\nThe proof uses summation by parts and the [[Cauchy criterion for series|06PVXMY0TH]].\nWith $b_n = (-1)^n$ it gives the [[alternating series test|0PK8QH1W1V]].\nIt also shows that $\\sum \\sin(n x) / n$ converges for each real $x$.\nMathlib states a form with a Cauchy sequence of partial sums, as `Monotone.cauchySeq_series_mul_of_tendsto_zero_of_bounded`.\n\n## History\n\nThe test is named after Peter Gustav Lejeune Dirichlet. It was published after his death, in 1862.\n\n## Explanations\n\n### L1 human\n\nThe partial sums of $\\sum b_n$ do not grow, and the weights $a_n$ shrink steadily to $0$.\nTogether these give enough cancellation for $\\sum a_n b_n$ to converge.\n",
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