Prove that $\lim \limits_{n\to \infty}\frac{n^{\alpha}}{(1+p)^n}=0$ Prove that if $p&gt;0$ and $\alpha\in \mathbb{R}$ then $$\lim \limits_{n\to \infty}\dfrac{n^{\alpha}}{(1+p)^n}=0.$$

No ideas how to prove it. Can anyone help please?

Prove without using logarithm.
Consider $\sum_{i=1}^\infty \frac{n^\alpha}{(1+p)^n}$, and perform the Ratio Test for convergence of this series:

$$\lim_{n\to \infty} \left| \frac{(n+1)^\alpha}{(1+p)^{n+1}} \cdot\frac{(1+p)^n}{n^\alpha}\right| = \lim_{n\to \infty}\left|\left(1 + \frac{1}{n}\right)^\alpha \frac{1}{1+p} \right| = \frac{1}{1+p} &lt; 1$$

As this limit is less than 1, the series is convergent, which is only possible if the sequence $\frac{n^\alpha}{(1+p)^n} \to 0$ as $n \to \infty$.
Take the logarithm and show that it goes to -infinity. If the logarithm goes to -infinity, then the sequence itself goes to 0.