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  "body": "\n## Statement\n\nLet $\\sum a_n$ be a real series that [[converges absolutely|0W3K3QPEWT]], with sum $A$.\nLet $\\sum b_n$ be a [[convergent|0CVVKY7GY2]] real series, with sum $B$.\nThen their [[Cauchy product|0B1K9PVWE9]] converges, and its sum is $A B$.\n\n## Remarks\n\nConvergence of both series is not enough. Take $a_n = b_n = (-1)^n / \\sqrt{n + 1}$. Both series converge, but $|c_n| \\ge 1$ for all $n$, so the Cauchy product diverges.\nFranz Mertens proved the theorem.\n\n## Explanations\n\n### L1 human\n\nOne series may converge only conditionally, if the other converges absolutely.\nThen the product of the two series is still the sum of their Cauchy product.\n",
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  "name": "Mertens' theorem on Cauchy products",
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