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What is region of convergence $(D\subset \mathbb C^2)$ of $$\sum_{n=0}^\infty(z_1^kz_2^l)^n$$ for fixed $k$ and $l$ integers. $z_1$ and $z_2$ are elements of complex plane.

What is method of finding the domain of convergence of several variable series.

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Thanks. Please write this comment as answer.... so that i can accept. – zapkm Apr 12 '12 at 9:02
up vote 1 down vote accepted

Here, the series is convergent if and only if $|z^k_1z^l_2|<1$, so $D=\{(z_1,z_2)\in\mathbb C^2:|z_1|^k\cdot |z_2|^l<1\}$. I don't whether there is a geometric interpretation, except for small values of $k$ and $l$.

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