What can we say about the convergence of this series $$\sum {z^n\over n!} $$

I used Alembert's Ration test and get $$\lim_{n \infty}{u_n\over u_{n+1}}={n+1\over z}$$ As this tends to $\infty&gt;1$ can i say that the given series is convergent for all values of $z$ ? 

Note : z is a complex number
Yes. A good thing to remind is the proof of this property, which in your case writes:

$$
\frac{u_{n+1}}{u_n} = \frac z{n+1}
\implies 
\left|\frac{u_{n+1}}{u_n}\right| \le 1/2 &lt;1
$$when $n$ is big enough, and then
$$
|u_n| \le C2^{-n}
$$for every $n$, for a certain $C&gt;0$. Hence the series is (absolutely) convergent.
The summation of the given series can be given as $e^z$ and hence the given series is convergent.