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  "body": "\n## Statement\n\nLet $\\sum a_n$ and $\\sum b_n$ be real series that [[converge absolutely|0W3K3QPEWT]], with sums $A$ and $B$.\nThen their [[Cauchy product|0B1K9PVWE9]] $\\sum c_n$ converges absolutely, and its sum is $A B$.\n\n## Remarks\n\nThe double series $\\sum_{i,j} a_i b_j$ converges absolutely, so its terms can be grouped in any order. The Cauchy product groups them by $i + j$.\n[[Mertens' theorem|0BJSWZJG18]] needs absolute convergence of only one of the two series, but then gives only convergence.\nMathlib states it for complete normed rings, as `tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm`.\n\n## Explanations\n\n### L1 human\n\nWhen both series converge absolutely, we may multiply them term by term as with finite sums.\nThe order in which we add the products does not change the result.\n",
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  "name": "Cauchy product of absolutely convergent series",
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