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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $n \\ge 1$.\nFor $\\alpha = (\\alpha_1, \\dots, \\alpha_n) \\in \\N^n$ write $X^\\alpha = X_1^{\\alpha_1} \\cdots X_n^{\\alpha_n}$.\nA *formal power series* in $X_1, \\dots, X_n$ over $R$ is a family $(a_\\alpha)_{\\alpha \\in \\N^n}$ of elements of $R$, written $\\sum_{\\alpha} a_\\alpha X^\\alpha$.\nThe set $R[\\![X_1, \\dots, X_n]\\!]$ of these series is a commutative ring with\n$$\\sum_\\alpha a_\\alpha X^\\alpha + \\sum_\\alpha b_\\alpha X^\\alpha = \\sum_\\alpha (a_\\alpha + b_\\alpha) X^\\alpha, \\qquad \\Big(\\sum_\\alpha a_\\alpha X^\\alpha\\Big)\\Big(\\sum_\\alpha b_\\alpha X^\\alpha\\Big) = \\sum_\\alpha \\Big(\\sum_{\\beta + \\gamma = \\alpha} a_\\beta b_\\gamma\\Big) X^\\alpha.$$\n\n## Remarks\n\nThe inner sum in the product is finite, since each $\\alpha$ has finitely many decompositions $\\alpha = \\beta + \\gamma$ in $\\N^n$.\nFor $n = 1$ this is the ring of [[formal power series|033WGN00BF]] in one variable, and $R[\\![X_1, \\dots, X_n]\\!] \\cong R[\\![X_1, \\dots, X_{n-1}]\\!][\\![X_n]\\!]$.\nExample: in $\\Z[\\![X, Y]\\!]$ the series $f = \\sum_{k \\ge 0} (X + Y)^k$ satisfies $(1 - X - Y) f = 1$, because the coefficient of each monomial of degree $k \\ge 1$ cancels between $(X + Y)^k$ and $(X + Y)(X + Y)^{k-1}$.\nNon-example: $\\sum_{k \\ge 0} X^{-k} Y^k$ is not a formal power series in $X, Y$, since the exponents must lie in $\\N^n$.\nA series is a [[unit|0XVEZDE9J8]] if and only if its constant coefficient $a_{(0, \\dots, 0)}$ is a unit of $R$.\n",
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