009C Sample Final 2, Problem 1

Test if the following sequences converge or diverge. Also find the limit of each convergent sequence.

(a)  $a_{n}={\frac {\ln(n)}{\ln(n+1)}}$

(b)  $a_{n}={\bigg (}{\frac {n}{n+1}}{\bigg )}^{n}$

Foundations:
L'Hopital'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:

(a)

Step 1:
First, we notice that  $\lim _{n\rightarrow \infty }{\frac {\ln(n)}{\ln(n+1)}}$  has the form  ${\frac {\infty }{\infty }}.$
So, we can use L'Hopital's Rule. To begin, we write
$\lim _{n\rightarrow \infty }{\frac {\ln(n)}{\ln(n+1)}}=\lim _{x\rightarrow \infty }{\frac {\ln(x)}{\ln(x+1)}}.$

Step 2:
Now, using L'Hopital's rule, we get
${\begin{array}{rcl}\displaystyle {\lim _{n\rightarrow \infty }{\frac {\ln(n)}{\ln(n+1)}}}&{\overset {L'H}{=}}&\displaystyle {\lim _{x\rightarrow \infty }{\frac {\frac {1}{x}}{\frac {1}{x+1}}}}\\&&\\&=&\displaystyle {\lim _{x\rightarrow \infty }{\frac {x+1}{x}}}\\&&\\&=&\displaystyle {\lim _{x\rightarrow \infty }1+{\frac {1}{x}}}\\&&\\&=&\displaystyle {1.}\end{array}}$

(b)

Step 1:

Step 2:

Final Answer:
(a)     $1$
(b)

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