Let $a\in [0,1)$. I want to show that $$\lim_{n\to \infty}{na^n}=0$$
My try : $$na^n={n\over e^{-(\log{a})n}}$$ and the limit is $${+\infty\over +\infty}$$ Hence by l'Hopital's rule we have that $$\lim_{n\to \infty}{1\over -(\log{a})e^{-(\log{a})n}}={1\over -\infty}=0$$
Is there any other way to compute this limit ? thanks!
(Similar) By L'Hopital's Rule $$\lim_{n \to +\infty}na^n=\lim_{n \to +\infty}\frac{n}{\frac{1}{a^n}}\overset{\frac{+\infty}{+\infty}}=\lim_{n \to +\infty}\frac{1}{{-\frac{1}{a^n}\ln(a)}}=-\frac{1}{\ln(a)}\lim_{n \to +\infty} a^n =0$$
Hint:
Another approach would be to look at the ratio of term $n+1$ to term $n$.
This gives $\frac{(n+1)a^{n+1}}{na^n}=\left(1+\frac{1}{n}\right)a$
Since $a<1$, this factor becomes strictly less than $1$ and then stays bounded away from $1$. Thus eventually the terms of the sequence start decreasing by these factors.
First, $a$ should be in $(0,1)$, otherwise L'Hopital's Rule won't work until the following step:
$\lim \limits_{n \to \infty} \frac{1}{-(\ln{a})e^{-n\ln{a}}}$
Second, we can use Squeeze Theorem to prove this limit.
Proof :
Let $a = \frac{1}{1+b}$, where $b>0$
$\because na^n > 0$
$na^n=\frac{n}{(1+b)^n}=\frac{n}{1+nb+\frac{n(n-1)}{2}b^2+...}$
$<\frac{n}{\frac{n(n-1)}{2}b^2}=\frac{2}{n-1}$
$\because \lim \limits_{n \to \infty} 0 = 0$
$\lim \limits_{n \to \infty} \frac{2}{n-1} = 0$
By Squeeze Theorem, $\lim \limits_{n \to \infty} na^n =0$
