First the intuition: the sequence grows way quicker than any polynomial. Hence the sequence diverges to .
Now for the proof. Because of the intuition, we try to bound from below, keeping the : for instance, for , (since ).
In a short while, you'll just have to say " is negligible compared to , hence ." But at that time, you may prefer a proof of this fact (it depends of what you know already).
So let . We prove that is larger than a divergent geometric sequence. For , . Because , we have thus . Hence there exists such that, for all , , i.e. . Hence, for , . Now you know that and (and ), so that .