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

Question

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

Possible duplicate of How to prove that exponential grows faster than polynomial? — Hans Lundmark, May 13 '18 at 07:55

score 3 · Accepted Answer · answered May 13 '18 at 04:39

3

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

answered May 13 '18 at 04:39

Trần Quang Minh

431

You could mention the "divergence test" to connect the dots. – mr_e_man May 13 '18 at 04:41

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

1 Answers1