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Let's consider the signal $x[n]=\{x[1],x[2],x[3]\}$

It's $\mathcal{Z}$ transform is $\frac {x[1]}{z}+\frac {x[2]}{z^2}+\frac {x[3]}{z^3}$

My textbook says it converges for all values of $z$ but there is a pole at $z=0$, so which one is true?

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When we speak about convergenge in all the $Z$ plane, the points $z=0$ and $z=\infty$ are not considered.

Note that by time-shifting the signal, you include/exclude zeros and poles at these 2 special points.

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