Domain of convergence for a Laurent series

Question

The Laurent series I have is given as $$\sum_{n=-\infty}^{-1} \frac{z^n}{|n-2|^3} + \sum_{n=0}^{\infty} \frac{z^n}{(2n)!}$$ I was trying to figure out the interval of convergence for the second half, I know how to do the first half, but on the second half, I tried the ratio test and the limit ends up going to $0$ which means the test is inconclusive, what can I do with the second half to find the radius of convergence?

Answer 1

For the second sum I get $r=\limsup_{n\to\infty}\dfrac1{\sqrt[n]{\dfrac1{2n!}}}=\infty$.

Since the first part converges for $|z|\gt1$, that will also be where the whole thing does.