Exponential Dominates Polynomial

Theorem
Let $\exp$ denote the exponential.

For any fixed $k \in \N$ and $\alpha > 0$ there exists $N \in \N$ such that $n^k < \exp(\alpha n)$ for all $n > N$.

Proof
Choose any $\displaystyle N > \frac{(k + 1)!}{\alpha^{k+1}}$, where $!$ denotes the factorial.

By Taylor Series Expansion for Exponential Function we have for any $n \in \N$:


 * $\displaystyle \exp(\alpha n) = \sum_{m \geq 0}\frac{(\alpha n)^m}{m!} > \frac{(\alpha n)^{k+1}}{(k+1)!}$

Therefore, for any $n > N$ we have:

This establishes the result.