Find limit $\lim_{n \to \infty }np^n , p \in (0,1)$

Question

I know that exponentials always grow faster than polynomials, but I ran into troubles when tried to prove this formally. Seems L'Hospital's rule doesn't work on it. Would you give me some directions on this?

Answer 1

L'Hopital's rule should work, once you realize that $np^n = ne^{n \ln p} = \frac{n}{e^{n \ln \frac{1}{p}}}$.

Answer 2

Hint: Let $p=\frac{1}{1+\delta}$. By the Binomial Theorem, if $n\gt 2$ we have $(1+\delta)^n\ge 1+n\delta+\frac{n(n-1)\delta^2}{2}\gt \frac{n(n-1)\delta^2}{2}$.

Answer 3

Consider the series $\sum_{n = 1}^{\infty}np^n$. Noticing that $\frac{(n + 1)p^{n + 1}}{np^n} \to p < 1$, we can apply the ratio test to conclude that the series converges. The upshot of this is that $np^n \to 0$.