009C Sample Midterm 1, Problem 1

Does the following sequence converge or diverge?

If the sequence converges, also find the limit of the sequence.

Be sure to jusify your answers!

$a_{n}={\frac {\ln n}{n}}$

Foundations:
L'Hôpital's Rule
Suppose that $\lim _{x\rightarrow \infty }f(x)$  and $\lim _{x\rightarrow \infty }g(x)$  are both zero or both $\pm \infty .$
If $\lim _{x\rightarrow \infty }{\frac {f'(x)}{g'(x)}}$  is finite or  $\pm \infty ,$
then $\lim _{x\rightarrow \infty }{\frac {f(x)}{g(x)}}\,=\,\lim _{x\rightarrow \infty }{\frac {f'(x)}{g'(x)}}.$

Solution:

Step 1:
First, we notice that
$\lim _{n\rightarrow \infty }\ln n=\infty$
and
$\lim _{n\rightarrow \infty }n=\infty .$
Therefore, the limit has the form ${\frac {\infty }{\infty }},$
which means we can use L'Hopital's Rule to calculate this limit.

Step 2:
First, we switch to the variable $x$ so we have functions and
can take derivatives. Thus, using L'Hopital's Rule, we have
${\begin{array}{rcl}\displaystyle {\lim _{n\rightarrow \infty }{\frac {\ln n}{n}}}&=&\displaystyle {\lim _{x\rightarrow \infty }{\frac {\ln x}{x}}}\\&&\\&{\overset {L'H}{=}}&\displaystyle {\lim _{x\rightarrow \infty }{\frac {{\big (}{\frac {1}{x}}{\big )}}{1}}}\\&&\\&=&\displaystyle {0.}\end{array}}$

Final Answer:
$0$

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