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  "body": "\n## Statement\n\nLet $R$ be a commutative ring, let $f, g \\in R[\\![X]\\!]$ be [[formal power series|072W67KFYK]], and let $n$ be a natural number.\nThen\n$$[X^n](f g) = \\sum_{i + j = n} [X^i] f \\cdot [X^j] g = \\sum_{i=0}^{n} [X^i] f \\cdot [X^{n-i}] g.$$\n\n## Remarks\n\nIn many texts this formula is the definition of the product of formal power series.\nThe sequence of coefficients of $fg$ is the convolution of the sequences of $f$ and $g$. This is the Cauchy product.\nExample: $[X^n] \\frac{1}{(1 - X)^2} = \\sum_{i=0}^{n} 1 \\cdot 1 = n + 1$.\n\n## Explanations\n\n### L1 human\n\nTo find the coefficient of $X^n$ in a product, collect all ways to get $X^n$ as $X^i$ times $X^j$.\nAdd the products of the matching coefficients.\n",
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