What is the limit of $\frac{n^k}{a^n}$

Question

What is $$\lim_{n\to\infty} \frac{n^k}{a^n},$$ where $a \in \mathbb{Q}, a>1,k\in\mathbb{N}$.

Answer 1

We know $\sum\limits_{n = 1}^{ + \infty } {\frac{{{n^k}}}{{{a^n}}}} < + \infty $ by ratio test. So we have $$\mathop {\lim }\limits_{n \to + \infty } \frac{{{n^k}}}{{{a^n}}} = 0.$$