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I have the following limit:

$$\lim_{n\rightarrow\infty}e^{-\alpha\sqrt{n}}\sum_{k=0}^{n-1}2^{-n-k} {{n-1+k}\choose k}\sum_{m=0}^{n-1-k}\frac{(\alpha\sqrt{n})^m}{m!}$$

where $\alpha>0$.

Evaluating this in Mathematica suggests that this converges, but I don't know how to evaluate it. Any help would be appreciated.

4 Answers 4