You have a mistake in your first series since when you reduced it from a doubly infinite to an infinite series, you forgot that the sign flips on the .

The calculation is actually:

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Your analysis on the radius of convergence is correct for the calculation you made. Now for this one that I've shown, what is the radius of convergence? I like your analysis in the attempt for the second part and as far as I can tell it has the right idea. Here is what I think:

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Now, the left sum only converges for |z|<1 and the right sum only converges if |z|>1, so the overal expression diverges for all z in the complex plane. The series doesn't converge at all, let alone to some function.

P.S. I notice you are a bit iffy with the indices of summation, so be careful.