I have the Laurent series
$$\sum_{n=1}^\infty \frac{n! z^n}{n^n} + \sum_{n=1}^\infty 2^n/z^n $$
and want to determine the annulus of convergence. The second sum clearly converges when |z|>2, but I can't quite get where the first one converges. Wolfram Alpha claims it converges on |z| < e by Ratio test but I don't see how they got this, I get L=1 (indeterminant result) when using ratio test.
If these are the two correct numbers, is the annulus of convergence then 2<|z|< e?