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  "body": "\n## Definition\n\nLet $R$ be a commutative ring and let $n$ be a natural number.\nThe coefficient of $X^n$ is the map $[X^n] : R[\\![X]\\!] \\to R$ that sends a [[formal power series|072W67KFYK]] $f = \\sum_{m \\ge 0} a_m X^m$ to $a_n$.\n\n## Remarks\n\nThe map $[X^n]$ is $R$-linear: $[X^n](f + g) = [X^n] f + [X^n] g$ and $[X^n](r f) = r\\, [X^n] f$.\nTwo series are equal if and only if all their coefficients are equal.\nThe coefficient of a product is given by the node [[product of power series|0H6KWAYS87]].\n\n## Explanations\n\n### L1 human\n\nThe coefficient $[X^n] f$ is the number in front of $X^n$ in the series $f$.\nFor $f = 1 + 2X + 3X^2 + \\dots$, the coefficient $[X^2] f$ is $3$.\n",
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