Limit to infinity of a sequence Considering the sequence $a_n = \frac{2^n}{n^4} $ I am to show in my analysis exercices that the $ \lim_{n\to\infty} a_n =\infty $ using the definition of the limit. Now so far my textbook has not really defined logarithms, although I certainly know how to use the basic rules to give that $2^n > n^5 $ when $ n>{\frac{5ln(n)}{ln2}}$ and then use the fact that $\frac{2^n}{n^4} > \frac{n^5}{n^4} $ which diverges to infinity . But I am not sure how to rigorously say when that statement is true without appealing to graphs, not to mention that as I said earlier I haven't really "defined" logarithms. So my next idea would be to take another expression that is smaller than  $a_n = \frac{2^n}{n^4} $ with converges to infinity but I am having trouble finding one. Any hints would be appreciated.

Using the binomial theorem, we can see that if $n\ge 5$ then 
$$(1+1)^n\gt \frac{n(n-1)(n-2)(n-3)(n-4)}{5!}.$$
If furthermore $n\ge 8$, then $(n)(n-1)(n-2)(n-3)(n-4)\gt (n^5)(2^4)$.  Thus for $n\ge 8$ we have
$$\frac{2^n}{n^4}\gt \frac{n}{(2^4)(5!)}.$$
