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

Question

Determine whether the sequence ${a n} \infty$
$n = 2$ with $a n = \frac{\ln}{n} n 2$ converges.

Show all steps.

Answer 1

Let us consider

$f (x) = \frac{\ln x}{x}$

where

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

for natural number

$n \geq 2.$

Now

We $lim_{n \to \infty} a n = lim_{x \to \infty} f (x)$

Therefore, using L'hospital we can say that

$lim_{n \to \infty} a n = lim_{n \to \infty} \times \frac{\ln n}{n} = \frac{\frac{lim_{n \to \infty} 1}{n}}{lim_{n \to \infty} 1} = 0$

$lim f (x) = \frac{lim (\ln^{'} x)}{1}$

Hence $lim_{x \to \infty} \times \frac{\ln x}{x} = 0$

Expert Answer