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 $n^k < \exp \left({\alpha n}\right)$ for all $n > N$.

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

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


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

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

This establishes the result.