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  "body": "\n## Summary\n\nA formal power series in $X_1, \\dots, X_n$ over a [[commutative ring|0310REJJY8]] $R$ is a possibly infinite sum $\\sum a_\\alpha X^\\alpha$ over all monomials $X^\\alpha$.\nNo convergence condition is imposed.\nThese series form the ring $R[\\![X_1, \\dots, X_n]\\!]$, which extends the one-variable [[formal power series|033WGN00BF]] ring and contains the [[polynomial ring|07GNX8GCYP]] $R[X_1, \\dots, X_n]$.\nOver a field, this ring is the basic example of a [[complete local ring|G9Q61B8A7C]] and of a regular local ring of dimension $n$.\n\n## Explanations\n\n### L1 human\n\nA polynomial in several variables is a finite sum of monomials such as $3x^2 y$.\nA formal power series in several variables allows infinitely many monomials, like $1 + x + y + x^2 + xy + y^2 + \\cdots$.\nYou add and multiply such series as you would polynomials, and every coefficient of a product only needs finitely many terms to compute.\nThe [[definition|KCBFS516DE]] makes this precise.\n",
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