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  "body": "\n## Definition\n\nLet $\\sum_{n=0}^{\\infty} a_n$ and $\\sum_{n=0}^{\\infty} b_n$ be real [[series|0JH4MN4K1C]].\nFor each $n$, let $c_n = \\sum_{k=0}^{n} a_k b_{n-k}$.\nThe series $\\sum_{n=0}^{\\infty} c_n$ is the Cauchy product of the two series.\n\n## Remarks\n\nThe term $c_n$ collects all products $a_i b_j$ with $i + j = n$.\nFor power series, the Cauchy product of $\\sum a_n x^n$ and $\\sum b_n x^n$ is $\\sum c_n x^n$. This is the rule for multiplying polynomials.\nThe Cauchy product of two convergent series need not converge. [[Mertens' theorem|0BJSWZJG18]] gives a sufficient condition.\n\n## Explanations\n\n### L1 human\n\nTo multiply two series, multiply every term of one by every term of the other.\nThe Cauchy product groups these products by the sum of their indices.\n",
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