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Let $a_{j,k}:=a_{j+2,k-1} \frac{(j+1)(j+2)}{k}$ for $k>0$ and $a_{j,0}:=(-1)^j$, thus $a_{j,k}=(-1)^j\frac{(2k+j)!}{j!k!}$, $j,k\geq 0$. Now, I have to show that the series defined by $S=\sum_{j,k}a_{j,k}x^j y^k$ diverges for any value except for $(x,y)=(0,0)$.

Probably, there is some analogous formula for the radius of convergence of a power series $S$ in multiple variables? Who can help?

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If $y=0$ it looks like a geometric series in $x$, which will converge for more than just $x=0$. Are you sure about the claim? – anon May 8 '12 at 20:40

Yes, the Cauchy Hadamard theorem generalizes to multidimensions, see here.

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