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  "body": "\n## Definition\n\nLet $R$ be a commutative ring.\nA formal power series in one variable $X$ over $R$ is a sequence $(a_n)_{n \\ge 0}$ of elements of $R$, written $\\sum_{n \\ge 0} a_n X^n$.\nThe set of these series is written $R[\\![X]\\!]$.\n\nAddition is termwise: $\\sum a_n X^n + \\sum b_n X^n = \\sum (a_n + b_n) X^n$.\nMultiplication is the Cauchy product: $\\left( \\sum a_n X^n \\right) \\left( \\sum b_n X^n \\right) = \\sum_n \\left( \\sum_{i+j=n} a_i b_j \\right) X^n$.\nWith these operations $R[\\![X]\\!]$ is a commutative ring, and it contains the polynomial ring $R[X]$.\n\n## Remarks\n\nThe word formal means that no convergence is required. A series is a sequence of coefficients, not a function.\nA series $\\sum a_n X^n$ is a unit in $R[\\![X]\\!]$ if and only if $a_0$ is a unit in $R$.\nThe [[coefficient|0M98XYVX6S]] of $X^n$ in a series $f$ is written $[X^n] f$.\n\n## Explanations\n\n### L1 human\n\nA formal power series is like a polynomial that may go on forever.\nYou add and multiply it like a polynomial, and you never substitute a number for $X$, so the question of convergence does not arise.\n",
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