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  "body": "\n## Summary\n\nThe region of convergence of a [[Z-transform|5FY0JQGHYZ]] is the set of complex numbers $z$ where its defining series converges absolutely.\nIt is an annulus centred at $0$, possibly empty, possibly with some boundary points.\nIt plays the part that the disc given by the [[radius of convergence|0N288P2TB5]] plays for a power series.\nA Z-transform is only known together with its region of convergence: two different sequences can have the same formula for $X(z)$ on two different regions.\nThe position of the region with respect to the unit circle tells whether a system is causal and stable.\n\n## Explanations\n\n### L1 human\n\nThe Z-transform is an infinite sum, and an infinite sum only gives a number when it converges.\nThe region of convergence is the set of points where it does.\nFor the sequence $1, 2, 4, 8, \\ldots$ the sum $\\sum 2^n z^{-n}$ converges exactly when $\\abs{z} \u003e 2$, the outside of a circle of radius $2$.\nFor a sequence that is nonzero only for negative $n$, the region is instead the inside of a circle.\n",
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