Exponential Dominates Polynomial

Theorem
Let $\exp$ denote the real exponential function.

For any fixed $k \in \N$ and $\alpha > 0$ there exists $N \in \N$ such that $x^k < \map \exp {\alpha x}$ for all real $x > N$.

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

By Taylor Series Expansion for Exponential Function we have for any $x \in \R_{\ge 0}$:


 * $\ds \map \exp {\alpha x} = \sum_{m \mathop \ge 0} \frac {\paren {\alpha x}^m}{m!} > \frac {\paren {\alpha x}^{k + 1} } {\paren {k + 1}!}$

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

This establishes the result.