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.

Hint: appealing to your approach, can you prove that $2^n > n^5$ for sufficiently large $n$?

 Suppose $2^{n}>n^{5}$ for some fixed $n$. Then, $2^{n+1}=2\cdot2^{n}>2\cdot n^{5}=(2^{1/5}\cdot n)^{5}=(n+an)^{5}$ where $a=2^{1/5}-1> 0$. Now, pick $n$ large enough so that $an>1$.

