# Determine whether the sequence {an}inftyn=2 with an =frac{ln}{n}n2converges. Show all steps.

Determine whether the sequence $\left\{an\right\}\mathrm{\infty }$
$n=2$ with $an=\frac{\mathrm{ln}}{n}n2$ converges.

Show all steps.

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Dora

Let us consider

$f\left(x\right)=\frac{\mathrm{ln}x}{x}$

where

$x\in R$ and $an=\frac{\mathrm{ln}n}{n}$

for natural number

$n\ge 2.$

Now

We $\underset{n\to \mathrm{\infty }}{lim}an=\underset{x\to \mathrm{\infty }}{lim}f\left(x\right)$

Therefore, using L'hospital we can say that

$\underset{n\to \mathrm{\infty }}{lim}an=\underset{n\to \mathrm{\infty }}{lim}×\frac{\mathrm{ln}n}{n}=\frac{\frac{\underset{n\to \mathrm{\infty }}{lim}1}{n}}{\underset{n\to \mathrm{\infty }}{lim}1}=0$

$limf\left(x\right)=\frac{lim\left({\mathrm{ln}}^{\prime }x\right)}{1}$

Hence $\underset{x\to \mathrm{\infty }}{lim}×\frac{\mathrm{ln}x}{x}=0$