Determine whether the series $\sum_{n=0}^\infty\frac{2^{n^2}}{n!}$ is convergent or divergent. 
Determine whether the series $$\sum_{n=0}^\infty\frac{2^{n^2}}{n!},$$ is convergent or divergent.

I know I have to use the ratio test.

Yes it diverges, and the simplest test to use is indeed the ratio test:
$$
\frac{a_{n+1}}{a_n}=\frac{2^{(n+1)^2} n!}{2^{n^2}(n+1)!}=\frac{2^{2n+1}}{n+1}\to \infty,
$$
as $n\to\infty$.
Hence the series diverges.
