Consider the series $$\sum_{k=0}^{\infty}\frac{1}{k!(z-k)^k}$$ Is there any domain of the complex plain where this series converges to a rational function?
Any hints will be appreciated
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1It appears that the sum will have poles at $k=1,2,\cdots$ – Martín-Blas Pérez Pinilla May 24 '14 at 07:23
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So we can conclude it doesn't converge to a rational function on any domain $\Omega\subseteq\Bbb{C}$ which contains an infinite subset of $\Bbb{N}$ – May 24 '14 at 08:55