Is the function an entire function? Is the following an entire function? (Here $z\in \mathbb{C}$)
$$\sum_{n=0}^\infty \frac{2^n}{n!}z^{3n}$$ 
($***$) So, here I first note that the function is a sum of powers of $z$. Now if I show that the sum converges for all $z\in \mathbb{C}$, the problem will be solved, right?
Using ratio test i get $\frac{a_{n+1}}{a_n}=\frac{2z}{n+1}$ which goes to $0$ as $n$ goes to $\infty$, showing that the function converges. The only problem is that is my statement $(***)$ correct?
Thanks in advance.

The function defined by the sum is entire, yes. For this, it suffices to check that the radius of convergence is infinite (which you have done); the result then follows as this implies that the power series is uniformly convergent everywhere.
