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   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm \infty ,}
then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\rightarrow \infty} \frac{f(x)}{g(x)}=\lim_{x\rightarrow \infty} \frac{f'(x)}{g'(x)}.}

Solution:

(a)

Step 1:

Step 2:

(b)

Step 1:

Step 2:

Final Answer:
(a)
(b)

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