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  "body": "\n## Definition\n\nLet $x = (x_n)_{n \\in \\Z}$ be a sequence of complex numbers, with [[Z-transform|5FY0JQGHYZ]] $X(z) = \\sum_{n \\in \\Z} x_n z^{-n}$.\nThe *region of convergence* of $X$ is the set\n$$\\mathrm{ROC}(x) = \\set{ z \\in \\C \\setminus \\set{0} : \\sum_{n \\in \\Z} \\abs{x_n} \\abs{z}^{-n} \u003c \\infty }.$$\n\n## Remarks\n\nLet $r = \\limsup_{n \\to \\infty} \\abs{x_n}^{1/n}$ and $1/R = \\limsup_{n \\to \\infty} \\abs{x_{-n}}^{1/n}$, with the conventions $1/0 = \\infty$ and $1/\\infty = 0$.\nBy the root test, the open [[annulus|0CW52NV5TC]] $r \u003c \\abs{z} \u003c R$ lies in the region, and no point with $\\abs{z} \u003c r$ or $\\abs{z} \u003e R$ does.\nThis is the [[radius of convergence|0N288P2TB5]] argument applied to each half of the [[Laurent series|0GXMPDNYX3]].\nPoints on the circles $\\abs{z} = r$ and $\\abs{z} = R$ may or may not belong to it.\nExample: $x_n = 1$ for $n \\ge 0$ and $x_n = 0$ for $n \u003c 0$ has region $\\abs{z} \u003e 1$; the point $z = 1$ fails because $\\sum_{n \\ge 0} 1$ diverges.\nNon-example: $z = 1/2$ is not in the region of $x_n = 2^n$ ($n \\ge 0$), since $\\sum 2^n 2^n$ diverges.\nSome texts add the point $z = \\infty$ to the region of a sequence with $x_n = 0$ for $n \u003c 0$, or $z = 0$ for a sequence with $x_n = 0$ for $n \u003e 0$.\n",
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