Solving a Complex power series combination

Question

Need help determining the circle and determining how this series converges and possibly the radius of convergence of this power series?

$$ \sum_{k=1}^{\infty}(-1)^{2k} \dfrac{(z-1)^k}{2i^{k}} \sum_{k=1}^{\infty}(-1)^{2k+1} \dfrac{(2z+1)^k}{2i^{k!}} $$

Not sure if I should try to combine them together into one-series and test using a convergence test or if something else would work?

Also it is defined on the complex plane if that helps.

Answer 1

This series never converges. In any case, the sequence $a_k$ must go to zero as $k\rightarrow \infty$. For the first series, this is true when $|z-1| < 1$. for the second series, this is true when $|2z+1| < 1$. These two regions never overlap so at least one of the series diverges for any value of $z$.